•*«crt    ••>-'  -• 


Engineering 
Library 


A  LABORATORY  MANUAL 

OF 

ALTERNATING  CURRENTS 


BY 


JOHN    H.    MORECROFT,    E.E. 


[BER    A.  I.  E.E. 

ASSISTANT    PROFESSOR    OF    ELECTRICAL    ENGINEERING 
COLUMBIA    UNIVERSITY 


LONGMANS,    GREEN,    AND    CO 

FOURTH  AVENUE  &  30TH  STREET,  NEW  YORK 

LONDON,  BOMBAY,  AND  CALCUTTA 
1912 


Engineering 
Library 


COPYRIGHT,  1912, 

BY 

LONGMANS,   GREEN   &  CO. 


Stanbopc  jpress 

H.GILSON  COMPANY 
BOSTON,  U.S.A. 


PREFACE. 


IN  composing  a  set  of  notes  intended  for  laboratory  use  the 
writer  has  to  consider  two  possible  methods  of  treating  the 
subject.  One  is  to  describe  explicitly  the  different  tests  to  be 
carried  out,  giving  diagrams  of  connections,  meters  to  be  used 
and  readings  to  be  taken,  and  even  giving  the  form  of  the  log 
for  keeping  the  laboratory  data.  Practically  nothing  is  said 
regarding  the  theory  involved  in  the  test,  the  reasons  for  taking 
the  various  measurements,  necessity  of  holding  certain  quanti- 
ties constant  and  allowing  others  to  vary,  etc.  This  method 
of  treatment  makes  the  performance  of  the  laboratory  work 
extremely  simple  for  the  student  as  well  as  for  the  instructor. 
It  is  probably  for  this  reason  that  most  laboratory  texts  are 
written  in  this  style.  Such  a  method  of  presenting  the  subject 
to  the  student,  however,  necessitates  very  little  thinking  on 
his  part.  The  connection  of  the  apparatus  and  taking  of  read- 
ings are  reduced  to  a  more  or  less  mechanical  operation,  and  even 
the  keen  student  does  not  get  from  his  laboratory  work  nearly 
as  much  material  for  study  as  he  should. 

The  second  method  of  handling  the  subject  matter  to  be  in- 
vestigated by  experiment  consists  in  a  careful  analysis  of  test 
to  be  performed,  of  the  different  variables  involved,  their  rela- 
tions to  one  another,  errors  likely  to  be  introduced,  etc.  As 
for  the  actual  performance  of  the  test,  the  student  should,  in  so 
far  as  is  feasible,  be  made  to  work  out  for  himself  the  scheme  of 
connections,  meters  to  be  used,  readings  to  be  taken,  arrange- 
ment of  data  on  the  record  sheet,  etc.  That  such  a  course  of 
analysis  of  the  test  to  be  made  is  the  proper  preparation  for  the 
student  to  have  before  performing  his  laboratory  work  is  evident 
when  it  is  remembered  that  the  practicing  engineer  must  do  just 
exactly  this  before  carrying  out  any  commercial  test. 

It  is,  of  course,  appreciated  that  the  lack  of  specific  directions 
in  a  laboratory  text  means  more  careful  preparation  by  the 
student  and  also  more  work  for  the  instructor,  but  as  a  labora- 
tory course  should  be  primarily  designed  to  teach  the  student 

ill 


254429 


iv  PREFACE 

methods  of  analysis,  and  to  emphasize  the  theory  presented  in 
lecture  courses,  rather  than  to  facilitate  the  perfunctory  per- 
formance of  a  set  of  experiments,  this  method  is  thought  to  be 
the  proper  one  and  so  is.  used  in  the  following  text. 

In  this  book,  intended  for  a  laboratory  manual,  it  would 
evidently  be  out  of  place  to  try  to  give  a  complete  mathematical 
analysis  of  all  alternating-current  phenomena.  The  various 
lecture  courses,  which  this  laboratory  course  is  supposed  to 
parallel,  will  take  up  the  general  theory  of  alternating  current 
circuits  and  machinery,  so  that  the  author  has  attempted  to  give 
here  only  those  elements  of  the  theory  which  apply  directly 
to  some  phenomenon  to  be  experimentally  investigated. 

The  analysis  of  the  action  of  alternating  current  meters  has  been 
given  more  in  detail  than  is  given  elsewhere;  it  has  seemed  to  the 
writer  that  the  student  should  thoroughly  appreciate  the  prin- 
ciples involved  in  the  construction  of  his  measuring  apparatus 
and  should  know  the  limitations  of  the  instruments;  too  often  He 
simply  reads  a  meter  and  takes  it  for  granted  that  the  indication 
has  a  meaning  which  he  understands;  with  distorted  waves, 
various  power  factors,  etc.,  there  are  many  intricate  points 
involved  which  affect  the  accuracy  of  a  meter  reading  and  it  is 
with  the  intention  of  calling  some  of  these  points  to  the  student's 
attention  (they  seldom  are  even  mentioned  in  the  various  books 
on  A.C.  theory)  that  mathematical  discussion  is  introduced  in 
several  places  and  that  some  experiments  are  given  to  illustrate 
the  points  covered  in  this  discussion.  The  experiments  serve 
also  to  bring  to  the  student's  notice  the  fact  that  our  alternat- 
ing current  standards  must  generally  be  referred  to  the  direct 
current  standards  and  the  basis  on  which  the  comparison  is  made. 

A  method  for  predicting  the  regulation  of  an  alternator  is 
given,  in  which  all  of  the  factors  entering  into  the  question  are 
logically  treated;  a  mathematical  discussion  of  armature  reaction 
in  single  and  polyphase  machines  is  given,  for  two  reasons  —  it  is 
not  generally  given  in  the  text-books  on  alternating-current  theory 
at  present  used,  and  the  rigid  proof  is  required  in  the  discussion 
of  armature  reaction  and  its  effect  on  alternator  regulation. 

The  experiments  described  are  designed  for  the  use  of  senior 
students  in  electrical  engineering,  and  so  considerable  knowledge 
of  the  laws  of  alternating  currents  is  assumed.  The  sequence 
of  the  tests  is  such  that  the  laboratory  work  will  parallel  and 
reinforce  the  lecture  courses  taken  by  the  student.  While  the 


PREFACE  V 

list  of  experiments  is  doubtless  incomplete,  it  will  serve  as  a  log- 
ical course  to  which  other  tests  may  be  readily  added  at  the 
discretion  of  the  individual  instructor,  to  suit  his  special  needs 
or  equipment.  Practically  all  of  the  tests  given  can  be  read- 
ily carried  out  with  the  equipment  of  the  average  laboratory. 

In  the  appendix  are  given  several  reproductions  from  ondograph 
or  oscillograph  records  which  serve  to  clear  up  certain  involved 
points  indicated  at  different  parts  of  the  text.  These  few  illustra- 
tions indicate  the  kind  of  work  for  which  the  curve  tracing  ap- 
paratus is  suited  and  represent  a  part  of  the  work  done  by  our 
senior  students  with  the  two  instruments  named.  It  is  the 
writer's  opinion  that  much  more  of  this  type  of  work  should  be 
incorporated  in  our  laboratory  courses. 

I  am  glad  to  express  my  thanks  to  Mr.  F.  L.  Mason,  of 
Columbia  University,  who  has  assisted  me  in  the  preparation  of 
the  book. 

J.  H.  M. 

COLUMBIA  UNIVERSITY, 
July,  1912. 


LIST   OF  EXPERIMENTS 


EXP.  PAGE 

1.  Wave  Forms  and  A.C.  Meters 1 

2.  The  Accuracy  of  A.C.   Meters  When   Measuring  an  A.C.   Wave 

Differing  Widely  from  the  Form  of  a  Sine  Wave 6 

3.  Power  and  Power  Factor  in  an  A.C.  Circuit 11 

4.  Measurement  of  Self -Induction  and  Effective  Resistance 18 

5.  Measurement  of  Coefficient  of  Mutual  Induction 20 

6.  The  Capacity  of  a  Condenser 22 

7.  Study  of  the  Reactions  in  an  Alternating-current  Circuit 25 

8.  Ohm's  Law  Applied  to  the  Alternating-current  Circuit 28 

9.  Circle  Diagram  for  Circuit  Containing  Resistance  and  Reactance. . .  33 

10.  Free  and  Forced  Vibrations;  Resonance  in  a  Circuit  Containing  Re- 

sistance, Inductance  and  Capacity 36 

11.  Magnetization  Curve  (No-load  Saturation  Curve)  of  an  Alternator 

and  External  Characteristic  on  Loads  of  Various  Power 
Factors 44 

12.  Full-load  Saturation  Curve,  Short-circuit  Current  (Synchronous- 

impedance  Curve)  and  Armature  Characteristic 47 

13.  Methods  for  Predetermining  the   External    Characteristic  of   an 

Alternator 53 

14.  Efficiency  of  an  Alternator  by  Rated  Motor 67 

15.   Efficiency  of  an  Alternator  by  the  "  Loss  "  Method 71 

6.   Parallel  Operation  of  Alternators 78 

17.  Study  of  the  Current  and  E.M.F.  Relations  in  Constant-potential 

Transformer  at  No-Load  and  at  Full  Load 95 

18.  Regulation  and  Efficiency  of  a  Transformer  by  Loading 102 

19.  Efficiency,  Regulation  and  Power  Factor  of  a  Transformer  by  the 

Loss  Method 104 

20.  Variation  of  Core  Losses  and  Exciting  Current  of  a  Transformer 

with  Varying  Impressed  E.M.F.  and  Frequency;  Separation 

of  Iron  Losses  into  Hysteresis  Loss  and  Eddy-current  Loss. .     113 

21.  Heat  Test  of  a  Transformer  by  Opposition  Method;  Polarity  Test..     116 

22.  Study  of  the  Constant-current  Transformer  and  Determination  of 

its  Characteristics 120 

23.  Parallel  Operation  of  Two  Constant-potential  Transformers 125 

24.  Two-phase  Power;  Uniformity  of  Power;  Different  Methods  of  Con- 

necting Circuits;  Vector  Addition  of  Currents  and  E.M.F.'s; 
Power  Factor 128 

25.  Current  and  E.M.F.  Relations  in  a  Three-phase  Circuit;  Power  and 

Power  Factor;  "  Equivalent  "  Resistance  and  Current 132 

26.  General  Polyphase    Transformation;    Two-phase  to   Three-phase 

Transformation  with  Balanced  and  Unbalanced  Load 138 

27.  Three-phase  Transformation;   Higher  Harmonics  in  Three-phase 

Circuits 143 

vii 


! 


Vlll  LIST  OF  EXPERIMENTS 

EXP.  PAOBJ 

28.  Phase  Characteristics  of  a  Synchronous  Motor;   Capacity  Action 

on  an  Inductive  Line 150 

29.  With  Motor  Excitation  Constant,  to  find  the  Relation  between 

Load  Current  and  Power  Factor;  Phase  Displacement  Varia- 
tions with  Load 158 

30.  Study  of  a  Rotary  Converter  Running  from  the  B.C.  End;  Voltage 

Ratios  for  Various  Numbers  of  Phases;  Variation  of  Voltage 
Ratio  with  Field  Strength;  External  Characteristic  for  In- 
ductive and  Noninductive  Loads;  Efficiency 162 

31.  Rotary  Converter  Running  from  A.C.  End;  Starting  by  Various 

Methods;  External  Characteristic,  with  and  without  Series 
Field  on  Inductive  Line  and  Noninductive  Line 168 

32.  Study  of  the  Auxiliary  Pole  Rotary  Converter;  Variation  of  Volt- 

age Ratio  with  Different  Field  Excitation  and  Examination 

of  Field  Form  with  Various  Voltage  Ratios 174 

33.  Study  of  the  Induction  Motor;  Its  Characteristics  by  Loading  with 

Prony  Brake  or  Generator 182 

34.  Prediction  of  Induction-motor  Characteristics  by  the  Method  of  the 

Circle  Diagram 192 

35.  The  Variable  Speed  Induction  Motor,  Its  Characteristics  by  Test 

and  Circle  Diagram;  Variation  of  Starting  Torque  with  Rotor 
Resistance 198 

36.  The  Characteristics  of  the  Single-phase  Induction  Motor;  Appli- 

cation of  Circle  Diagram  for  Predetermination;  Starting  as 

a  Repulsion  Motor 202 

37.  Study  of  the  Induction  Generator,  Magnetization  Curve;  External 

Characteristic  when  Excited  by  Synchronous  Motor;  Change 
from  Motor  Action  to  Generator  Action  with  Variation  of 
Speed  when  Connected  to  Power  Line  of  Constant  Fre- 
quency    206 

38.  The  Single-phase  Series  Motor 216 

39.  The  Mercury  Arc  Rectifier 222 

APPENDIX,  with  illustrations  of  special  problems  solved  by  use  of  the 

ondograph  or  oscillograph 231 

PLATES    1-  4.    Current  forms  in  oscillating  circuits. 

PLATE      5.         Armature  reaction  in  single-phase  alternator. 

PLATE      7.         Circulating  current  between  two  alternators. 

PLATE      7.         Voltage  forms  and  exciting  current  of  a  transformer. 

PLATE      8.         Current  taken  by  a  transformer  when  first  connected  to 

line  of  normal  voltage. 
PLATES    $-14.   Voltage  and  current  relations  in  transformers  connected 

to  three-phase  line,  showing  the  "  wabbling  neutral." 
PLATE  15.  Current  forms  in  a  synchronous  motor  with  various  field 

excitations. 

PLATES  16-18.   Armature  reaction  in  rotary  converters. 
PLATES  19-23.   Current  forms  in  the  various  coils  of  a  rotary  converter. 
PLATES  24-26.   Field  forms  of  an  auxiliary  pole  rotary  converter. 
PLATE    27.         Voltage  forms  between  various  taps  of  the  armature  of 

an  auxiliary  pole  rotary  converter. 


ALTEENATING    CUEEENTS 


EXPERIMENT  I. 
WAVE  FORMS  AND  A.C.  METERS. 

ALL  ordinary  A.C.  theory  is  worked  out  on  the  supposition 
that  the  voltages  and  currents  to  be  considered  are  simple  sine 
functions  of  time.  It  is,  therefore,  important  to  see  how  nearly 
this  condition  actually  obtains.  The  fundamental  connection 
between  direct  current  and  alternating  current  units  is  founded 
on  the  fact  that  the  alternating  current  ampere  flowing  through 
a  given  resistance  produces  heat  at  the  same  rate  as  would  the 
direct  current  ampere  through  the  same  resistance.  Now  the  rate 
at  which  heat  is  produced  in  any  conductor  is  given  by  the 
formula,  Heat  =  PR. 

In  a  continuous  current  circuit  this  heat  is  generated  at  a 
uniform  rate  because  7  is  a  constant.  In  an  alternating  current 
circuit  this  is  not  true  because  the  current  varies  in  magnitude 
with  respect  to  time.  Calling  the  instantaneous  value  of  the  alter- 
nating current  i,  the  maximum  value  7m,  we  have  the  equation, 
i  =  Im  sin  ojt. 

The  heat  generated  by  such  a  variable  current  can  be  expressed 
only  by  an  integration,  which  is 

Heat  =   /  i?Rdt,  where  R  is  the  resistance  of  the  conductor 

in  which  the  heat  is  being  generated.  The  "  effective  "  value  of 
the  alternating  current  is  defined  as  that  constant  value  of 
current,  which,  flowing  through  the  same  resistance,  will  pro- 
duce in  a  time  equal  to  one  cycle  of  the  alternating  current,  the 
same  heat  as  is  produced  by  the  varying  current  throughout  one 
cycle. 

Calling  such  effective  value  7,  we  must  have 

PRt  =    I   i2Rdt,  or  when  R  is  a  constant, 

Jo 

Pt  =    I    iz  dt,  the  time  involved  in   the   integration 

«/o 

being  that  necessary  for  one  complete  cycle  of  current  values. 

1 


2  ALTERNATING  CURRENTS 

This  equation  defines  /  in  terms  of  i  and  evidently  gives  the 
relation,  P  =  average  i2,  or  /  =  A/average  value  of  i2.  In  the 
same  way  is  obtained  E  =  A/mean  square  of  e,  where  E  and  e 
are  effective  and  instantaneous  values  of  voltage  in  the  alternat- 
ing current  circuits. 

Now  an  A.C.  meter  is  always  calibrated  in  terms  of  effective 
values  and  it  is  to  investigate  the  validity  of  the  above  formulas 
that  part  of  this  test  is  designed. 

Some  scheme  is  necessary  for  obtaining  wave  forms  of  current 
and  voltage.  Such  an  instrument  as  the  oscillograph  or  ondo- 
graph  would  draw  the  desired  wave  forms,  but  a  much  more 
accurate  method  is  that  known  as  the  "  point  by  point  "  method. 
This  method  consists  in  balancing  the  instantaneous  value  of 
alternating  E.M.F.  against  a  known  continuous  E.M.F.  The 
apparatus  required  is  a  potentiometer  drum,  telephone  receiver 
and  a  revolving  switch  on  the  shaft  of  the  alternator  supplying 
the  waves  to  be  investigated.  The  revolving  switch  is  a  disc 
of  insulating  material  having  imbedded  in  its  periphery  a  metal 
strip  which  is  connected  to  the  alternator  shaft  on  which  the 
disc  is  mounted. 

In  Fig.  1,  there  is  shown  only  one  metal  strip  in  the  disc. 
This  scheme  will  give  only  as  many  pulses  per  minute  through  the 


A.C. 


D.G. 


telephone  as  the  alternator  is  turning  revolutions  per  minute.  As 
the  ear  does  not  detect  easily  tones  of  lower  than  perhaps  sixty 
vibrations  per  second,  especially  in  a  laboratory  where  there  are 
many  other  noises  of  higher  tone,  the  disc  would  have  to  run 
3600  r.p.m.  for  satisfactory  use.  But  there  may  be  more  than 
one  strip  placed  in  the  disc;  in  general  there  should  be  one  strip 
for  every  360°  (electrical)  of  the  armature.  If  the  alternator 


WAVE  FORMS  3 

has  four  poles  there  should  be  two  strips  diametrically  opposite; 
with  an  eight-pole  machine  there  should  be  four  strips,  90° 
(mechanical)  apart.  The  disc,  shown  in  Fig.  1,  and  the  descrip- 
tion of  its  operation,  which  follows,  is  supposed  to  be  for  a  two- 
pole  generator.  A  brush  mounted  on  the  alternator  frame 
(but  insulated  from  it)  makes  contact  with  the  metal  strip  at 
the  same  time  in  each  revolution  of  the  armature.  As  the 
E.M.F.  of  the  generator  alternates  synchronously  with  the  revo- 
lution of  the  armature,  it  is  evident  that  such  a  disc  furnishes 
a  means  of  closing  a  circuit  at  the  same  phase  of  each  successive 
E.M.F.  or  current  wave  furnished  by  the  alternator. 

The  scheme  of  connections  is  as  given  in  Fig.  1.  A  resistance 
H  serves  to  give  a  variable  source  of  continuous  E.M.F.,  the 
contact  F  being  movable  along  H.  The  alternating  E.M.F.,  the 
wave  form  of  which  is  to  be  measured,  is  connected  through  the 
reversing  switch  E  to  the  local  circuit  consisting  of  the  telephone 
receiver  C,  revolving  disc  B,  and  part  of  the  resistance  H.  When 
the  metal  strip  D  makes  contact  with  the  brush  G,  there  will  be 
acting  in  this  local  circuit  two  E.M.F. 's,  the  instantaneous  value 
of  the  alternating  E.M.F.  and  the  continuous  E.M.F.  from  the 
potentiometer.  The  sliding  contact  F  may  be  moved  until  the 
voltage  from  the  potentiometer  just  balances  the  instantaneous 
value  of  the  alternating  E.M.F.,  when  no  current  will  flow  through 
the  local  circuit  at  the  closing  of  the  switch  B,  and  so  no  noise  will 
be  heard  in  the  telephone.  (It  may  be  necessary  to  reverse  switch 
E  to  bring  about  the  balance.  Why?)  When  a  balance  is  ob- 
tained the  D.C.  voltmeter  A  is  read  and  so  the  instantaneous 
value  of  the  alternating  E.M.F.  is  obtained.  By  moving  the 
brush  G  through  360  electrical  degrees  and  obtaining  successive 
balances  at  a  sufficient  number  of  points,  the  wave  of  alternating 
voltage  may  be  plotted  through  an  entire  cycle.  (A  high  resist- 
ance R  is  used  to  prevent  overheating  the  telephone  when  the 
local  circuit  is  badly  unbalanced.) 

Instead  of  connecting  the  local  circuit  to  the  blades  of  switch 
E,  it  may  be  connected  to  a  two-point  jack  which  can  be  con- 
nected to  several  receptacles,  and  balances  obtained  for  several 
different  E.M.F.  waves  for  the  same  setting  of  the  brush  G.  So 
not  only  forms,  but  relative  phases  of  current  and  E.M.F.'s  may 
be  obtained.  It  should  be  noted  that  the  effective  value  of  any 
alternating  E.M.F.  to  be  measured  must  not  be  greater  than  0.707 
of  the  D.C.  voltage  across  the  potentiometer  H.  Why? 


4  ALTERNATING  CURRENTS 

In  any  A.C.  measuring  instrument  (with  the  exception  of 
polyphase  meters)  the  impelling  force  on  the  moving  system 
is  a  varying  quantity,  while  the  resisting  force  (as  that  of  a 
spring)  is  constant  for  a  given  position  of  the  moving  system. 

For  a  given  position  of  the  moving  system  the  following 
relation  must  hold: 

time  integral  of  varying  force  =  time  integral  of  constant  force. 

Of  course,  at  any  given  instant  one  of  the  forces  will  be  greater  or 
less  than  the  other  so  that  actually  the  resultant  force  on  the 
moving  system  is  generally  not  equal  to  zero  and  the  system 
begins  to  move.  But  as  the  resultant  force  will  vary  with  twice 
the  frequency  of  the  impelling  force  (giving  frequency  of  resultant 
force  between  perhaps  50  and  250  per  second)  it  is  evident  that 
the  moving  system  cannot  oscillate  with  any  appreciable  ampli- 
tude and  so  will  assume  what  seems  to  be  a  stationary  position 
such  that  the  above  equation  is  true. 

The  curves  of  alternating  current  and  E.M.F.  are  to  be  obtained 
by  the  " point  by  point"  method,  balancing  the  instantaneous 
value  of  the  alternating  E.M.F.  against  a  variable,  known,  con- 
tinuous E.M.F.  The  curve  of  alternating  current  is  obtained  by 
finding  the  form  of  the  "fall  of  potential"  curve  over  a  known, 
noninductive  resistance  through  which  the  alternating  current  is 
passed.  Any  ordinate  of  this  curve,  divided  by  the  value  of  the 
known  resistance,  gives  the  corresponding  instantaneous  value  of 
the  current. 

Make  connections  as  given  in  Fig.  2,  and  take  a  reading  for 
every  15°  (electrical).  For  a  noninductive  resistance  a  lamp 

A.C.  ammeter 


Non-inductive  |  |    S 

resistance    I  I  g     || 


FIG.  2. 

bank  or  noninductive  rheostat  may  be  used,  the  resistance  of 
which  is  to  be  measured  with  direct  current.  Plot  the  in- 
stantaneous A.C.  values,  using  time  for  the  abscissae.  Square 
each  value  obtained  and  construct  curve  of  "  squared  ordinates." 


WAVE  FORMS  5 

By  planimeter,  or  by  counting  squares  of  section  paper,  or  by 
graphical  integration,  obtain  area  of  this  squared  curve,  divide 
by  its  base  and  so  get  the  "  mean  square  "  of  the  A.C.  wave. 
Get  the  square  root  of  this  value  and  compare  with  reading  of 
A.C,  meter. 

Obtain  in  a  similar  way  the  "  average "  value  of  the  A.C. 
wave  and  calculate  its  "  form  factor."  (This  form  factor  is 
1.11  for  sine  waves.) 

With  same  base  and  on  same  section  paper  construct  a  sine 
curve  having  same  amplitude  as  measured  A.C.  wave.  This 
construction  will  show  how  nearly  the  A.C.  wave  used  approaches 
a  sine  curve  and  will  account  for  any  discrepancy  in  the  form 
factor. 

Keep  voltage  and  resistance  constant  throughout  test. 


EXPERIMENT  II. 

ALTERNATING  CURRENT  METERS  ON  CIRCUITS  OF  DISTORTED 

WAVE  FORM. 

THE  dynamometer  and  hot-wire  types  of  meter  will  accurately 
record  the  mean  square,  whatever  the  shape  of  the  wave  form, 
but  other  types  of  A.C.  meters  will  not  generally  do  so.  The 
moving  iron-vane  type  of  ammeter  (e.g.,  Weston)  will  only  have 
a  force  proportional  to  the  (current)2  so  long  as  the  iron  vane 
maintains  constant  permeability.  By  correctly  designing  the 
meter  this  type  will  give  quite  accurate  results  on  waves  very 
much  distorted. 

A  meter  whose  impelling  force,  due  to  the  current  flowing 
through  its  windings,  varies  directly  with  the  first  power  of  the 
current,  if  calibrated  with  sine  wave  current,  will  not  read  ac- 
curately when  used  with  a  wave  form  differing  from  sine  form. 
The  magnitude  of  its  error  will  be  measured  by  the  ratio  of  the 
wave  form  factor  to  the  form  factor  of  a  sine  wave. 

Now  if  the  current  being  measured  is  a  distorted  wave  it  may 
be  represented  by  a  fundamental  sine  wave  and  a  series  of 


FIG.  3. 

harmonics  of  various  amplitudes.  If  the  impelling  force  of  the 
meter  is  proportional  to  the  square  of  the  current,  the  reading 
of  the  meter  will  be  equal  to  A/Ai2  +  A22  +  Af  +,  etc.,  where 
AI,  A2)  Az,  etc.,  represent  the  amplitudes  (effective  values),  of 
the  fundamental  and  the  various  harmonics. 

Consider  the  case  of  an  A.C.  ammeter  recording  the  exciting 
current  of  a  transformer,  the  meter  being  one,  the  impelling 
force  of  which  is  proportional  to  the  square  of  the  instanta- 

6 


METER  ACCURACY  WITH  DISTORTED  WAVE  7 

neous  value  of  the  current  flowing  through  it.  This  wave  is 
much  distorted  but  may  be  fairly  represented  by  the  equation, 
x  =  AI  cos  co£  +  AS  cos  (3  ut  +  0),  where 

AI  =  amplitude  of  fundamental, 

co  =  2  TT  X  fundamental  frequency, 
As  =  amplitude  of  third  harmonic, 

0  =  distance  on  X  axis  between  the  zero  values  of  the  two 
waves,  as  in  Fig.  3. 

Now  the  meter  reading,  with  such  a  current,  will  be  propor- 
tional to  the  average  value  of  the  impelling  force,  or  average 
value  of  x2. 

Average  force  =  -  /    (Ai  cos  at  +  ^.3  cos  (3  ut  +  0))2  dt 

TT  Jo 

1  r*  i   r* 

7Tt/o  71"  *sQ 

+  -  I2A1A3  cos  coZ  cos  (3  coZ  +  0)  dt. 

7T  JQ 

Ai2 

The  values  of  the  first  two  integrals  are  quite  evidently  -»-  and 

A    2 

-~- »  respectively. 

The   third  integral  can  be  obtained  by   expanding  the  term 
cos  (3  coZ +  0). 
The  third  integral 


7T 

2 


A  A     C* 

-  I    cos  ut  (cos  3  coi  cos  0  —  sin  3  ut  sin  0)  dt. 

7T          JQ 

=  K   I    (cos  3  ut  cos  co£)  dt  —  KI  I     (sm3utcosut)dt; 
Jo  Jo 

where  the  K's  are  constants,  involving  AI,  A3  and  functions  of  0. 
We  may  put  cos  3  wt  =  cos  2  co£  cos  wt  —  sin  2  ut  sin  coi 

=  (cos2  ut  —  sin2  coi)  cos  cot  —  2  sin2  co£  cos  co£, 
therefore, 

r(cos3cufcoscuO^=<  /    cos4co^-3/    si 
ft/O  t/0 

=  [cos3  ut  sin  CO^]Q  =  0. 


8  ALTERNATING  CURRENTS 

In  the  same  way 

r(sin  3  ut  cos  ut)  dt  =  I    (3  sin  ut  cos3  coZ)  dt  —  I    (sin3  cot  cos  co£)  dt 
«/o  «/o 

-  -[J  cos*  ««];;-[}  sin*  «l];=0. 

Therefore,  the  force  acting  on  the  moving  element  of  the  instru- 
ment =  K  l~-  +  -~  \  and  as  the  meter  scale  is  graduated  in 
terms  of  the  square  root  of  the  impelling  force,  the  reading  of  the 

meter  will  be 

But  -  -  =  (—7")  =  (effective  value)2  of  fundamental  current, 

Aa2 

and  —  =  (effective  value)2  of  third  harmonic  current, 

and  the  current  indicated  by  the  ammeter  will  be  I  = 

where  I\  and  Is  are  the  effective  values  of  the  fundamental  and 

harmonic. 

This  demonstration  may  be  easily  generalized  and  it  is  found 
that  the  meter  indication  for  any  complex  wave  is 


+  ^2  +  /52  +     '    •    '     In2    •     •     • 

As  the  circuit  of  the  A.C.  voltmeter  is  practically  a  nonin- 
ductive  resistance  the  current  flowing  through  its  moving  element 
will  be  of  exactly  the  same  shape  as  the  E.M.F.  wave.  There- 
fore, when  acted  upon  by  a  complex  wave  the  meter  indication 


will  be  E  =  V  E] 

where  E\  —  amplitude  of  fundamental,  etc. 

The  validity  of  these  two  formula  may  be  readily  tested. 
Connect  two  alternators,  of  different  frequencies,  in  series  with 
one  another.  Read  the  voltage  of  each  alternator  and  of  the 
line,  and  results  will  be  obtained  as  indicated  in  Fig.  4.  Or  two 
currents  of  different  frequencies,  the  amplitude  of  each  of  which 
can  be  measured,  may  be  used  to  check  the  current  formula. 
Send  both  currents  through  the  same  ammeter  and  it  will  be 
found  that  the  meter  reading  will  be  the  square  root  of  the  sum 
of  the  squares  of  the  individual  currents. 

In  electrical  quantities  the  different  frequencies  are  all  simply 
related,  i.e.,  there  are  the  3rd,  5th  or  7th  harmonics,  etc.  In 


ioo  volts 


METER  ACCURACY  WITH  DISTORTED  WAVE  9 

case  the  two  frequencies  are  very  nearly  alike,  the  integral  of  the 
cross  product  will  have  different  values  depending  upon  the 
interval  over  which  the  integral  is  taken.  The  force  acting 
on  the  moving  element  of  the  meter  will  vary  with  a  compara- 
tively slow  period  and  if  the  frequencies  are  close  enough  to- 
gether the  reading  of  the  meter 
will  be  a  fluctuating  one,  the 
amount  of  fluctuation  depend- 
ing  upon  the  relative  magni-  /\ioo  volts 

j          -    ,,  c 

tude  of  the  two  frequencies. 

In  Fig.  4,  if  the  frequencies 

were  60  and  60.  1  ,  respectively,  FIG.  4. 

and   the    voltages    each    100 

volts,  the  voltage  of  the  line  would  oscillate  between  0  and  200. 

As  it  is  easier  to  obtain  a  distorted  wave  of  current  than  of 
E.M.F.,  ammeters  will  be  used  for  this  test.  To  obtain  the  dis- 
torted wave  of  current  use  the  exciting  current  of  a  transformer, 
operating  at  about  25  per  cent  above  normal  voltage.  Even 
though  a  sine  wave  of  E.M.F.  be  applied  to  a  transformer,  the 
exciting  current  will  be  much  distorted  (for  reasons  to  be  dis- 
cussed later,  see  p.  108,  paragraph  beginning,  "  The  current 
which  flows,"  etc.). 

The  value  of  resistance  inserted  in  series  with  the  transformer 
must  not  be  high,  otherwise  the  current  wave  will  not  be  much 


To"  point  by  pomt"  apparatus 

.AAAA/VN.               /1\ 

iNoninductive    ^-^               o 
,^,•0*0™^  Iron  vane      o 

§ 

A.C. 

meter          o 

g 

supply 

o 

° 

I 

0 
x—  N.                             ° 

P 

Dynamometer  or 

Hotwire  meter 

FIG.  5. 

distorted.  If  the  drop  of  potential  over  the  resistance  is  not 
more  than  10  per  cent  of  the  voltage  impressed  on  the  trans- 
former, the  distorted  wave  will  be  easily  obtained,  but  if  much 
more  than  this  amount  of  resistance  is  inserted  in  the  circuit 
the  amount  of  current  distortion  obtainable  will  be  much 


10  ALTERNATING  CURRENTS 

reduced.  The  reason  for  this  will  be  given  in  a  following  experi- 
ment, dealing  with  the  wave  forms  of  transformer  quantities. 

With  connections  as  shown  in  Fig.  5  take  a  set  of  readings 
similar  to  those  taken  in  Experiment  1.  Use  a  Siemen's  dyna- 
mometer or  hot  wire  meter,  for  one  meter  and  an  iron-vane  am- 
meter for  the  other. 

Construct  curve  of  current  from  instantaneous  values  obtained 
and  get  Vmean  square  by  the  method  described  in  Experiment  1 ; 
compare  these  values  with  the  indications  of  the  meters ;  calculate 
the  form  factor  of  the  wave. 

Why  should  some  types  of  A.C.  meters  not  record  accurately 
when  measuring  distorted  wave  forms? 

Caution.  —  After  adjusting  the  voltage  of  the  alternator  to 
about  the  right  value,  with  the  transformer  circuit  open,  open 
the  field  switch  of  the  alternator,  close  the  transformer  circuit, 
and  then  close  the  alternator  field  circuit.  If  this  method  of  pro- 
cedure is  not  used  the  ammeters  are  likely  to  be  injured. 


EXPERIMENT  III. 
POWER  AND  POWER  FACTOR  IN  AN  A.C.  CIRCUIT. 

IN  a  circuit  through  which  an  alternating  current  is  flowing 
the  current  and  pressure  are  generally  not  in  the  same  phase. 
The  power  in  any  circuit  is  the  integral  of  the  product  of  in- 
stantaneous values  of  E  and  /  and  this  product,  when  E  and  / 
are  out  of  phase,  will  be  negative  during  a  part  of  the  cycle. 
During  the  negative  part  of  the  power  cycle,  current  is  flowing 
in  a  direction  opposite  to  the  impressed  E.M.F.,  i.e.,  the  circuit 
is  feeding  power  back  into  the  supply  circuit. 

When  E  and  I  are  represented  by  rotating  vectors  it  is  readily 
seen  that  the  power  will  be  given  by  the  expression  El  cos  0, 
where  0  is  the  phase  difference  of  E  and  7. 

Because  of  the  occurrence  of  the  negative  loops  in  the  power 
cycle,  power  is  generally  a  double-frequency  function.  The  only 
case  when  this  is  not  true  is  when  the  current  and  impressed 
E.M.F.  are  exactly  in  phase,  in  which  case  the  negative  loop  dis- 
appears and  the  power  is  expressed  by  the  product  EL  The 
indicating  wattmeter,  in  the  same  way  as  mentioned  in  Experi- 
ment 1,  balances  an  oscillating  force  against  a  steady  force,  and 
the  oscillating  force  is  sometimes  negative  (when  cos  0  <  1). 
The  moving  system  of  the  wattmeter  tries  to  vibrate  with  the 
frequency  of  the  power  curve,  but,  owing  to  its  inertia  and  the 
rapidity  of  change  of  the  impressed  force,  it  cannot  oscillate,  and 
so  assumes  some  intermediate  position. 

The  current  through  the  potential  coil  of  the  wattmeter  should 
be  in  phase  with  the  E.M.F.  of  the  circuit  being  measured. 
This  means  that  the  potential  circuit  must  be  wound  non- 
inductively.  In  case  a  multiplier  is  used  with  a  wattmeter, 
care  should  be  taken  to  see  that  it  is  so  wound.  The  slight 
amount  of  inductance  always  present  in  the  potential  circuit 
introduces  an  error  varying  with  the  power  factor  of  the  circuit 
being  tested;  it  is  inappreciable  on  large  power  factors  but  be- 
comes much  larger  as  the  power  factor  decreases.  This  effect 
is  shown  in  Fig.  6,  where  the  error  caused  by  angle  6  (due  to 
inductance  in  potential  circuit)  is  seen  to  be  large  when  <j>  is 

11 


12 


ALTERNATING  CURRENTS 


tan0 


large  and  comparatively  small  when  <£  is  small.  When  0  is  small 
the  phase  relations  of  E  and  I  are  as  shown.  Representing  the 
current  by  01  and  the  true  phase  of  the  voltage  by  OE,  then, 

owing  to  the  inductance  of  the 
D  TV  potential  circuit,  the  wattmeter 

acts  as  if  the  impressed  E.M.F. 

was  not  OE,  but  OE' ,  lagging 

behind  OEby  the  angle  0,  where 

Reactance  of  potential  coil 
Resistance  of  potential  coil 

The  wattmeter  should  indi- 
cate the  power,  OA  X07,  where- 
as it  really  indicates  OA'  X  07. 
The  percentage  error  is  given 

FlG-  6-  by  the  fraction^- 

When  the  power  factor  of  the  circuit  to  which  the  wattmeter 
is  connected  is  small,  conditions  are  as  indicated  by  OD  and  07)'. 

r>  r>/ 

The  percentage  error  is  now  -77^-,  which  is  much  larger  than  was 

Un 

the  case  with  the  load  of  power  factor  =  cos  <£. 
Expressed  analytically  we  have, 

True  power          =  El  cos  <£, 

Indicated  power  =  El  cos  (<£  —  0)  =  El  {cos  0  cos  0  +  sin<£  sin0J . 

As  0  is  always  small,  cos  0  does  not  appreciably  differ  from  unity 
and  we  have 

Indicated  power  =  El  (cos  4>  +  sin<£  sin0), 

which  shows  that  the  error  introduced  varies  with  the  sine  of  the 
characteristic  angle  of  the  circuit  being  measured.  The  indicated 
power  is  too  large  when  the  circuit  being  measured  has  a  lagging 
current  and  too  small  when  the  current  in  the  circuit  leads  the 
E.M.F. 

When  the  question  of  power  measurement  in  a  circuit,  having 
either  the  current  or  E.M.F.,  or  both,  complex  quantities,  is  con- 
sidered, the  action  of  the  wattmeter  must  be  analyzed. 

The  field  in  such  an  instrument  is  produced  by  the  current  in 
the  circuit,  and  the  current  in  the  moving  coil  is  of  the  same 
shape  as,  and  proportional  to,  the  voltage  of  the  circuit  being 


POWER  AND  POWER  FACTOR  13 

tested.      The  impelling  force  will  then  vary  as  the  product  of 
these  two  quantities. 

Suppose  the  E.M.F.  is  simple  harmonic 

-  e  =  Em  cos  (at,  and  the  current  is  complex, 
x  =  AI  cos  (cat  +  <£)  +  Az  cos  (3  at  +  B). 

What  will  the  wattmeter  read  when  connected  to  such  a  circuit? 
Average  Force 

=  -  /    Emcos(atlA1cos((at  +  (f>)  +  AzCOs(Z(at  +  6)ldt 

7T«/o 

=  -  /    EmAi  cos  (at  cos  ((at  H-  <f>)  dt 

TT  JQ 

+  -  I     EmA3  cos  wt  cos  (3  ait  +  B)  dt. 

TT  Jo 

Now,  it  has  previously  been  shown  that  an  integral  of  the  form 
of  the  second  term  is  equal  to  zero,  therefore 

Average  Force  =  -  I     EmAi  cos  (at  cos  ((at  +  <£) 

TT  JQ 


This  analysis  may  be  carried  out  for  a  current  containing  any 
number  of  harmonics,  but  it  will  be  found  that  the  wattmeter 
reading  is  equal  to  the  product  of  the  effective  values  of  the 
E.M.F.  and  the  fundamental  current  and  the  cosine  of  their  phase 
difference.  So  long  as  one  of  the  quantities,  E.M.F.,  or  current, 
is  simple  harmonic,  the  wattmeter  reading  is  entirely  indepen- 
dent of  any  upper  harmonics  which  may  exist  in  the  other 
quantity. 

If,  however,  there  is  a  third  harmonic  in  both  E.M.F.  and 
current,  then  the  wattmeter  reading  will  be  proportional  to 


-  / 

TfJo 


cos  <o£  +  #3  cos  (3  co£  +  7)  j  \(AiGOs((at 


Upon  evaluation  this  integral  yields  two  terms  : 

V      A  V      A 

Wattmeter  reading  =  —i  —^  cos  0  H  —  -i  —^=  cos  (7  —  0), 


14  ALTERNATING  CURRENTS 

that  is,  the  wattmeter  reading  is  the  sum  of  the  watts  obtained 
by  multiplying  the  effective  value  of  each  voltage  by  the  effective 
value  of  the  current  of  the  same  frequency  by  the  cosine  of  their 
phase  difference. 

From  this  simple  discussion  it  is  evident  why  the  power 
factor  of  a  circuit  upon  which  there  is  impressed  a  simple  sine 
curve  of  E.M.F.,  and  in  which  there  is  flowing  a  complex  current, 
is  more  or  less  a  fictitious  quantity.  The  ratio  of  watts  to  (volts 
X  amperes)  gives  a  certain  number,  but  the  significance  of  such 
number  is  not  at  once  apparent.  The  wattmeter  takes  no 
account  of  the  upper  harmonics,  but  the  product,  volts  X  am- 
peres, does  involve  them  indirectly  because  the  ammeter  reads 
the  (root  mean  square)  of  the  amplitudes  of  the  fundamental 
and  all  upper  harmonics  which  may  be  present.  The  value  of 
0  so  obtained  does  not  signify  the  phase  difference  of  the  zero 
points  of  the  E.M.F.  and  current  waves  of  the  circuit.  Neither 
does  it  signify  the  phase  difference  of  the  E.M.F.  and  the  fun- 
damental current  wave.  The  real  significance  of  0  so  obtained 
is  the  phase  displacement  of  the  E.M.F.  and  a  fictitious  simple 
harmonic  current,  the  effective  value  of  which  is  the  same  as 
that  of  the  complex  wave,  the  frequency  of  which  is  the  same 
as  that  of  the  fundamental  current,  and  the  phase  of  which  is 
such  that  it  would  produce  on  the  wattmeter  the  same  effect 
as  is  produced  by  the  actual  complex  wave. 

When  accuracy  of  measurement  is  desired  care  must  be 
exercised  in  connecting  the  different  instruments,  as  the  errors 
introduced  by  the  power  consumption  of  the  meters  themselves 
may  be  appreciable.  An  A.C.  voltmeter  may  use  as  much  as 
0.1  ampere  or  more.  If  the  total  current  which  the  circuit  is 
supplying  is  1  ampere,  the  voltmeter  may,  if  improperly  con- 
nected, introduce  an  error  of  the  order  of  10  per  cent. 

Suppose  a  circuit  as  shown  by  full  lines  in  Fig.  7,  and  that  the 
current  taken  by  the  circuit  H  is  1  ampere  and  that  the  voltmeter 
takes  0.1  ampere.  The  ammeter  will  indicate  the  vector  sum  of 
the  current  through  H  and  that  through  the  voltmeter.  Even 
though  the  magnitude  of  the  voltmeter  current  is  known,  the  rela- 
tive phase  of  the  circuit  current  and  voltmeter  current  is  not  gen- 
erally known,  so  that  the  ammeter  reading  cannot  be  corrected. 

Now,  if  the  voltmeter  is  so  connected  that  its  current  does  not 
flow  through  the  ammeter  (as  shown  by  the  dotted  line  in  Fig.  7), 
the  reading  of  A  will  be  the  true  current  through  H.  But  now 


POWER  AND  POWER  FACTOR 


15 


the  voltmeter  reading  is  not  the  voltage  across  circuit  H,  but  some- 
thing greater  because  there  must  be  some  drop  of  potential 
through  the  ammeter.  This  drop  will  generally  be  small  and  in 
most  circuits  would  probably  be  inappreciable.  If  the  current 
through  H  is  very  large  compared  to  the  voltmeter  current  the 
connections  may  be  made  as  shown  in  the  full  lines  of  Fig.  7  and 
the  error  of  measurement  will  be  small. 

When  a  wattmeter  is  used  for  measuring  the  power  in  a  circuit 
the  same  precautions  must  be  observed  in  connecting  its  potential 
coil  as  were  noted  above  for  the  voltmeter  connection.  If  the 
wattmeter  is  connected,  as  shown  in  the  full  lines  of  Fig.  8,  the 


/=?       1  ampere 


H 


FIG.   7. 


FIG.  8. 


reading  of  the  meter  represents  the  power  used  in  coil  H  plus  the 
power  used  in  the  potential  coil  itself.  If  this  loss  is  known  it 
may  be  subtracted  from  the  wattmeter  reading;  if 

E  =  potential  difference  applied  to  potential  coil 
R  =  resistance  of  potential  coil,  then  the 

correction    (amount  to    be   subtracted   from   wattmeter   read- 

.     ,       W- 
mg)  =  -g- 

In  case  the  wattmeter  is  of  the  compensated  type  no  error  of 
this  sort  is  incurred  and  so  no  correction  need  be  applied.  In 
this  type  of  meter  the  coil  generating  the  magnetic  field,  in  which 
the  moving  coil  lies,  and  with  which  the  moving  coil  reacts  to 
produce  deflection  of  the  meter,  is  wound  of  two  wires  side  by 
side,  one  of  them  of  large  enough  cross  section  to  carry  the  rated 
current  capacity  of  the  meter  and  the  other  of  fine  wire  through 
which  the  potential-circuit  current  flows.  The  internal  connec- 
tions of  the  coils  are  as  shown  in  Fig.  9.  If  the  meter  is  connected 
to  the  circuit,  as  shown  in  the  full  lines  of  Fig.  8,  it  is  evident  that 
the  M.M.F.  due  to  coil  A  is  produced  by  the  current  7  +  i,  where 
/  =  current  through  circuit  H,  i  =  current  through  potential 
circuit  of  meter.  Now  the  compensating  winding  is  so  con- 


16 


ALTERNATING  CURRENTS 


High  resistance 


nected  that  when  the  meter  is  properly  connected  to  circuit  H, 
i.e.,  in  such  fashion  that  the  meter  gives  a  positive  deflection, 
the  current  i  through  the  compensating  winding  produces  a 
M.M.F.  in  direction  opposite  to  that  produced  by  coil  A.  The 

resultant  M.M.F.  and  hence  the 
field  influencing  the  moving  coil 
C,  is  due  to  current  { (7  +  i)  —  i  { 
or  /.  The  reading  of  the  watt- 
meter is,  therefore,  independent 
of  the  current  taken  by  the  po- 
tential coil  and  the  meter  should 
be  connected  as  shown  in  the  full 
lines  of  Fig.  8.  If  the  potential 

circuit  is  connected  as  shown  by  the  dotted  line  of  Fig.  8  the 
wattmeter  will  indicate  inaccurately,  as  the  field  of  the  meter  will 
be  proportional  to  (7  —  i)  instead  of  I. 

With  connections  given  in  Fig.  10,  keeping  voltage  and  fre- 
quency of  supply  constant,  obtain  the  curves  of  current  and 
voltage  by  the  "  point  by  point  "  method.  Read  also  the  am- 
meter, voltmeter  and  wattmeter.  Adjust  the  circuit  so  that 


Compensating  winding 

FIG.  9. 


FIG.  10. 

cos  cf>  is  0.8  or  less.  Take  one  set  of  curves  using  an  induc- 
tance coil  having  an  air  core,  in  which  case  the  current  will  be  a 
sine  curve  and  the  measured  phase  difference,  as  obtained  from 
curve  sheet,  should  check  with  value  calculated  from  the  ratio, 

watts      ,_, 

Then  take  a  set  of  curves  using  an  iron-core 


cos 


-,j 


inductance,  forcing  through  it  enough  current  to  thoroughly  satu- 
rate the  iron  core,  under  which  condition  the  current  curve  will 
be  much  distorted.  The  measured  0  from  the  curves  will  not 
now  check  with  the  value  obtained  from  the  meters.  This  fact 
emphasizes  the  necessity  of  confining  the  simpler  calculations 
of  A.C.  quantities  to  sinusoidal  functions. 


POWER  AND  POWER  FACTOR  17 

With  the  data  obtained,  construct  curves  of  current  and  volt- 
age and  from  these  two  construct  the  curve  of  power.  Get  the 
area  of  the  power  curve  (by  planimeter  or  otherwise)  and  com- 
pare with  the  wattmeter  reading.  Scale  off  the  angular  dis- 
placement between  zero  points  of  current  and  E.M.F.  waves; 
find  the  cosine  of  this  angle  and  compare  with  the  power  factor 
obtained  by  the  ordinary  formula. 


EXPERIMENT  IV. 

MEASUREMENT  OF  SELF-INDUCTION  AND  EFFECTIVE 
RESISTANCE. 

WHENEVER  the  current  through  a  conductor  is  varied  a  counter 
E.M.F.  is  set  up  in  the  conductor  due  to  the  change  in  strength 
of  the  magnetic  field  surrounding  it.  The  magnitude  of  this 
C.E.M.F.  depends  upon  the  constants  of  the  circuit  (number  of 
turns  and  the  reluctance  of  the  magnetic  circuit)  and  upon  the 
rate  of  change  of  the  current. 

The  C.E.M.F.  may  be  expressed  by  the  equation, 

C.E.M.F.  =  (constant)  X  (^\  ; 

this  constant  for  any  given  circuit  is  called  the  coefficient  of  self- 
induction,  generally  designated  by  L.  It  is  to  be  noted  that  L 
is  not  a  constant  when  the  reluctance  of  the  magnetic  circuit  is 
variable,  as  e.g.,  when  there  is  iron  in  the  magnetic  circuit,  in 
which  case  L  will  depend  upon  the  value  of  the  current.  From 
this  it  will  be  noted  that  L  will  vary  throughout  the  A.C.  cycle, 
because  the  permeability  of  the  iron  varies  with  the  instantane- 
ous value  of  the  alternating  current.  L  is,  however,  ordinarily 
obtained  in  terms  of  effective  E  and  7,  in  which  case  it  is  treated 
as  a  constant  throughout  the  cycle. 

If  there  were  no  resistance  in  the  circuit  considered  we  might 
write: 

Impressed  voltage  =  L  -r  - 

As  all  circuits  have  resistance  this  equation  must  include  the 
resistance  drop,  so  we  have 

Impressed  voltage  —L-^  +  Ri. 

di 

The  fact  that  i  is  a  sine  function  of  time  and  that  -r.  is  a  cosine 

at 

function  shows  that  the  two  components  of  the  C.E.M.F.  must 
be  added  as  vectors  at  right  angles  to  give  the  impressed  E.M.F. 
as  a  vector. 

18 


SELF-INDUCTION  AND  EFFECTIVE  RESISTANCE          19 

Considering  the  vector  diagram  and  using  the  ordinary  nota- 
tion we  have  E  =  I  V(2irfL)2  +  R*,  where  E  and  /  are  effective 
values. 

The  power  consumption  in  the  circuit  is  independent  of  the  in- 
ductance component  of  the  C.E.M.F.  as  this  component  is  at  right 
angles  to  the  phase  of  the  current,  hence  the  resistance  component, 
IR,  must  be  such  a  quantity  that,  when  multiplied  by  the  cur- 
rent, it  gives  the  total  power  consumption  in  the  circuit.  When 
there  is  iron  in  the  magnetic  circuit,  power  will  be  used  up  due 
to  hysteresis  and  eddy  currents  in  the  iron.  Hence  the  power 
consumption  will  be  greater  than  I2r  (r  being  the  ohmic  resist- 
ance of  the  circuit)  and  when  we  write 

Power  consumed 


R  stands  for  something  more  than  ohmic>esistance,  and  is  called 
the  "effective"  resistance  of  the  circuit;  to  get  this  effective 
resistance  of  any  circuit  measure  the  power  in  watts  by  a  watt- 
meter and  divide  by  the  square  of  the  current.  This  effective 
resistance  will  generally  be  greater  than  the  ohmic  resistance; 
it  may,  however,  be  less  than  the  ohmic  resistance  when  two 
mutually  inductive  circuits  are  considered,  moving  with  respect 
to  one  another,  as  in  the  case  of  the  induction  generator. 

Measure  the  effective  resistance  and  coefficient  of  self-induc- 
tion of  two  inductance  coils  (one  having  air  core  and  one  with 
magnetic  circuit  of  iron)  at  several  values  of  current  and  fre- 
quency. Account  for  changes  in  their  values?  Measure  the 
ohmic  resistance  of  the  coil  and  compare  with  the  effective 
resistance.  Calculate  the  power  factor  of  the  circuit. 

With  a  given  impressed  voltage  on  the  coil  would  you  expect 
L  and  R  to  vary  with  the  frequency.  Why? 


EXPERIMENT  V. 
MEASUREMENT   OF   COEFFICIENT   OF   MUTUAL   INDUCTION. 

WHENEVER  the  magnetic  flux  through  a  circuit  varies  there  will 
be  an  E.M.F.  set  up  in  the  circuit.  The  flux  may  be  generated 
by  current  in  the  coil  considered  or  by  another  coil  so  situated 
with  respect  to  the  one  considered  that  part  of  its  flux  threads 
the  second  circuit.  It  is  then  clear  that  whenever  the  current 
in  the  first  coil  is  varied,  thereby  changing  the  strength  of  its 
magnetic  field,  there  will  be  an  E.M.F.  set  up  in  the  second  coil. 
The  magnitude  of  this  E.M.F.  will  depend  upon  the  number  of 
turns  in  the  two  coils,  the  position  of  one  with  respect  to  the 
other,  reluctance  of  path  of  the  mutual  flux  and  the  rate  of 
change  of  current  in  the  first.  The  value  of  the  E.M.F.  generated 

di 
in  the  second  coil  may  be  written,  e  =  M  -r ,  where  i  is  the  current 

in  the  first  coil  and  M  is  called  the  coefficient  of  mutual  induction 
of  the  two  coils. 

When  the  reluctance  of  the  path  of  the  mutual  flux  is  a  con- 
stant quantity,  then  the  coefficient  M  will  be  a  constant  for  all 
values  of  current,  and  will  be  the  same,  whichever  coil  is  used 
for  creating  the  magnetic  field.  If  the  magnetic  circuit  is  com- 
posed in  part  or  altogether  of  iron,  and  the  two  coils  have  dif- 
ferent numbers  of  turns,  then  the  value  of  M  determined  will  be 
in  general  different  when  each  coil  in  turn  is  used  to  create  the 
field;  but,  if,  when  both  measurements  are  taken,  the  field  is 
at  the  same  density  then  M  will  be  the  same  in  both  cases.  It  is 
self-evident  that  if  we  desire  M  to  be  as  large  as  possible  the  two 
coils  should  be  so  situated  that  all  the  field  produced  by  the  first 
coil  threads  the  second.  This  is  never  possible  but  is  approxi- 
mated in  the  constant  potential  transformer,  where  the  two  coils 
are  generally  either  concentric  or  laminated,  the  sections  of  the 
two  coils  being  interspersed. 

If  a  transformer  is  desired  having  a  variable  M  (as  in  the  case 
of  the  constant  current  transformer)  then  a  leakage  path  of 
variable  reluctance  is  used,  which  prevents  more  or  less  of  the 
field  of  the  first  coil  from  linking  with  the  second.  In  measuring 

20 


MUTUAL  INDUCTION  21 

M,  effective  values  of  E.M.F.  and  current  are  read,  in  terms  of 
which  the  equation  for  induced  E.M.F.  becomes 

E.M.F.i  = 


As  the  coefficients  of  self-induction  of  the  two  coils  and  their 
coefficient  of  mutual  induction  all  depend  upon  the  field  strength 
and  number  of  turns  of  the  coils  it  is  evident  that  some  mathe- 
matical relation  must  be  obtainable,  which  will  express  one  in 
terms  of  the  others. 

If  n\  =  number  of  turns  in  coil  No.  1, 

nz  =  number  of  turns  in  coil  No.  2, 
<£i  =  flux  per  ampere,  through  coil  No.  1,  produced  by 

current  in  No.  1, 
02  =  flux  per  ampere,  through  coil  No.  2,  produced  by 

current  in  No.  2, 

K  =  coefficient  of  leakage  between  the  two  coils,  i.e.,  the 
fraction  by  which  0i,  e.g.,  must  be  multiplied  to 
give  the  flux  through  coil  No.  2  due  to  current 
in  coil  No.  1,  or  vice  versa, 
Li  =  tti0i, 
LZ  =  ^202, 
M  = 

LiZ/2   = 


_ 

M2  =  n!tt20i02#2  or  M  =  K  VLiL2. 


So  the  coefficient  of  leakage  between  the  two  coils  may  be  deter- 
mined by  measuring  M,  LI  and  L2. 

Using  first  two  separate  coils  with  air  cores,  find  out  how  M 
varies  with  current,  frequency  and  relative  position  of  coils. 

Then  use  two  coils  on  the  same  iron  circuit  and  make  similar 
tests.  If  possible,  introduce  a  better  leakage  path  between  the 
two  coils  and  again  measure  M. 


EXPERIMENT   VI. 
TO  MEASURE  THE  CAPACITY  OF  A  CONDENSER. 

THE  most  natural  method  to  employ  in  this  test  would  be  to 
measure  the  two  quantities  by  which  capacity  is  defined.  If  we 
measure  the  charge  Q,  required  to  bring  the  condenser  plates  to 

a  difference  of  potential  V,  then  by  definition,  C  =  ^  •     This 

method  is  used  where  C  is  small  and  a  ballistic  galvanometer  is 
available  for  measuring  Q.  In  case  C  is  as  large  as  a  few  micro- 
farads or  more,  a  convenient  way  of  measuring  it  is  to  determine 
the  charging  current  when  a  known  alternating  E.M.F.  is  applied 
to  its  terminals. 

If  we  apply  a  varying  E.M.F.  to  the  condenser  the  fundamental 
equation  becomes 


dV 
When  -rr  =  2  TrfEm  sin  2  irft,  then  as  i  must  be  a  function  similar 

dV 
to  -j-,  we  have 

i  =  Im  sin  2  irftj 
and  C  2  irfEm  sin  2  irft  =  Im  sin  2  irft  or 

in  effective  values  2irfCE  =  I;  hence,  if  we  measure  E  and  7, 
C  can  be  calculated  when  /  is  known. 

If  an  accurate  determination  of  C  is  desired  an  electrostatic 
voltmeter  of  low  capacity  must  be  employed  to  measure  E.  If 
a  closed-circuit  voltmeter  is  used,  one  of  two  errors  may  be 
introduced.  Reading  the  charging  current  when  the  voltmeter 
is  shunted  across  the  condenser  may  give  too  large  a  current  due 
to  that  taken  by  the  voltmeter  in  parallel  with  the  condenser, 
or  may  give  too  small  a  value  if  the  voltmeter  has  an  appreciable 
inductance.*  An  inductance  and  a  condenser  in  parallel  with 
each  other  may  be  so  proportioned  that  practically  all  of  the 

*  This  condition,  of  course,  never  occurs  with  an  ordinary  commercial  meter. 

22 


MEASUREMENT  OF  CAPACITY  23 

charging  current  for  the  condenser  is  furnished  by  the  inductance 
and  will  not  be  read  by  an  ammeter  in  the  supply  circuit. 

A  perfect  condenser  would  use  no  power,  but  in  practice  the 
dielectrics  are  imperfect  and  the  continually  reversing  polariza- 
tion causes  an  energy  loss  similar  to  the  hysteresis  loss  in  iron 
subjected  to  magnetic  reversals.  There  is  also  more  or  less 
mechanical  vibration  of  the  condenser  plates  which  necessitates 
power  consumption;  also  there  occurs  actual  leakage  of  current 
from  one  plate  of  the  condenser  to  the  other  and  this  leakage 
causes  energy  loss  in  the  condenser  dielectric.  This  dielectric 
loss  is  generally  very  small,  so  that  the  charging  current  will  lead 
the  impressed  voltage  by  an  angle  of  between  85°  and  90°.  The 
dielectric  loss  will  heat  a  condenser  and  care  should  be  exercised 
that  a  condenser  intended  for  temporary  use  (starting  condenser 
for  single-phase  induction  motor,  e.g.)  is  not  left  connected  to 
the  circuit  longer  than  necessary.  Also  a  condenser  should  not 
be  subjected  to  a  higher  voltage  than  that  for  which  it  was 
designed,  as  the  increased  losses  may  cause  the  dielectric  to 
deteriorate  if  not  actually  to  puncture.  In  case  the  dielectric 
used  in  the  condenser  is  paraffine,  the  danger  from  overheating 
the  condenser  is  very  evident.  The  electrical  resistance  of 
paraffine  decreases  enormously  as  its  temperature  rises;  its 
resistance  at  90°C.,  e.g.,  is  only  one-fortieth  of  the  value  at  40°C.; 
if  the  first  heating  is  due  to  actual  leakage  current  through  the 
dielectric,  the  condenser  is  almost  sure  to  break  down  if  left  on 
the  circuit  for  a  sufficient  length  of  time,  as  the  heating  effect  is 
cumulative;  an  increased  temperature  produces  an  increase  in 
the  rate  at  which  power  is  being  used  in  the  dielectric  and  so  the 
temperature  increases  until  the  dielectric  is  punctured. 

Owing  to  the  fact  that  a  dielectric  such  as  paraffine  has  a  lag 
angle  (i.e.,  its  polarization  lags  somewhat  behind  the  polarizing 
force)  a  slight  decrease  in  capacity  may  be  expected  as  the  fre- 
quency increases,  because  the  charge  does  not  have  sufficient 
time  to  "  soak  in  "  or  effect  the  same  degree  of  polarization  as 
it  would  if  the  E.M.F.  was  applied  continuously  in  the  same 
direction.  Certain  paraffine  condensers  showed  a  decrease  from 
38.5  m.f.  to  35.8  m.f.  as  the  frequency  was  increased  from  40 
cycles  to  80  cycles. 

When  the  dielectric  used  has  a  small  lag  angle  then  very  little 
power  is  used  in  charging  and  discharging  the  condenser.  Such 
is  the  case  with  mica.  Paraffine  paper  condensers  give,  however, 


24 


ALTERNATING  CURRENTS 


considerable  loss,  certain  condensers  tested  giving  results  as 
shown  in  Fig.  11.  It  will  be  observed  that  the  loss  follows 
somewhat  the  same  variations  as  would  the  hysteresis  loss  in  a 
piece  of  iron  subjected  to  various  m.m.f.'s  and  frequencies. 


I    .5 

s  •< 

f   .3 

.2 

1 


120  volts 


100  volta 


80  volta 


volts 


20          30          40          50          60          70          80 
Frequency  in  cycles  per  Second 

FIG.  11. 

By  means  of  an  A.C.  ammeter  and  voltmeter  determine  the 
capacity  of  a  condenser  and  make  proper  tests  to  see  if  the 
capacity  changes  with  frequency  and  voltage.  By  use  of  a 
wattmeter  determine  the  power  used  in  the  condenser  at  various 
voltages  and  frequencies  and  calculate  the  phase  angle  of  the 
charging  current.  The  wattmeter  reading  must  be  corrected  for 
the  error  incurred  by  the  inductance  of  its  potential  circuit,  if 
accurate  results  are  to  be  obtained.  This  error  is  explained  in 
Experiment  3. 


EXPERIMENT   VII. 

STUDY  OF  THE  REACTIONS  IN  AN  ALTERNATING- 
CURRENT   CIRCUIT.* 

IN  a  direct-current  circuit  containing  nothing  but  a  resistance 
the  only  reaction  opposing  the  impressed  E.M.F.  is  the  IR  drop, 
and  so  we  have  the  reaction  equation  in  the  familiar  form  of 
Ohm's  law,  i.e.,  E  =  IR. 

If  we  have  a  circuit  containing,  besides  resistance,  a  counter 
E.M.F.,  such  as  the  circuit  of  a  storage  battery  being  charged, 
then  the  reaction  equation  must  be  written  in  the  form 

E  =  IR  +  -E'counter, 

or  in  its  general  form,  known  as  Kirchhoff's  second  law,  it  is 
E  =  2  IR  +  S  counter  E.M.F.'s. 

It  should  be  noticed  that  this  method  of  analyzing  the  problem 
of  the  electric  circuit,  i.e.,  equating  the  impressed  force  to  the 
sum  of  the  reacting  forces,  is  much  more  logical  than  the  one 
usually  employed  in  which  the  current  flowing  is  put  equal  to 
the  impressed  force  divided  by  the  resistance;  it  should  be  borne 
in  mind  that  the  quantities  which  can  be  measured  experiment- 
ally are  reactions.  The  resistance  of  a  wire  cannot  be  measured 
directly,  but  the  quantity  measured  is  the  drop  in  pressure, 
or  resistance  reaction,  caused  by  current  flowing  through  the 
conductor. 

In  a  Wheatstone  bridge  the  actual  resistance  of  the  unknown 
conductor  is  not  measured.  A  current  is  sent  through  the  con- 
ductor and  the  bridge  is  so  adjusted  that  the  resistance  reaction 
of  the  unknown  resistance  is  equal,  or  proportional,  to  the  re- 
action due  to  the  same  current  passing  through  a  conductor 
of  known  resistance.  In  measuring  resistance  by  the  "  fall  of 

*  The  idea  of  attacking  all  A.C.  problems  by  means  of  the  fundamental 
principle  that,  in  any  electrical  circuit,  the  impressed  force  must  be  equal  to 
the  sum  of  all  the  reactions  in  the  circuit,  I  owe  to  Prof.  M.  I.  Pupin.  I  have 
become  convinced  of  the  utility  of  the  reaction  conception  in  solving  electri- 
cal problems  by  actually  using  it  in  the  solution  of  some  original  problems  on 
which  I  have  been  recently  working  with  Prof.  Pupin. 

25 


26  ALTERNATING  CURRENTS 

potential  "  method,  using  voltmeter  and  ammeter,  the  actual 
resistance  is  not  measured.  The  resistance  reaction  is  the 
quantity  measured. 

In  the  alternating-current  circuit  containing  resistance,  in- 
ductance and  capacity  in  series  there  are  three  reactions  to 
consider. 

The  resistance  drop,  iR,  is  at  every  instant  proportional  to 
the  current,  and,  as  it  opposes  the  flow  of  current,  it  is  180°  out 
of  phase  with  the  current.  The  quantity  R  will  not  be  the 
actual  ohmic  resistance  of  the  circuit  but  will  be  the  "  effective  " 
resistance  as  described  in  Experiment  4. 

The  inductance  reaction  of  the  circuit  is  proportional  to  the 
time  rate  of  change  of  the  current  and  is  given  by  the  expression 

di 

—  L  -7-  -     From  consideration  of  a  sine  wave  it  is  evident  that 
at 

when  the  current  is  zero  and  increasing  the  inductance  reaction 
will  be  a  negative  maximum,  i.e.,  it  lags  behind  the  current  by 
90°.  Hence  the  component  of  the  impressed  E.M.F.  which  must 
be  used  in  overcoming  this  reacting  force  will  lead  the  current 
by  90°. 

The  magnitude  of  the  counter  E.M.F.  of  the  capacity  is  given 
by  the  formula  (Experiment  6) 

dV 

dt       C 


•^   or  V  =  £  /  i  dt. 


When  the  action  of  the  condenser  is  analyzed  it  is  seen  that  this 
reacting  E.M.F.  has  its  maximum  negative  value  at  the  end  of  a 
positive  alternation  of  the  current.  This  is  evident  because  the 
condenser  pressure  acts  to  decrease  its  charge,  and  at  the  end  of 
a  positive  alternation  of  current  the  charge  will  be  a  positive 
maximum.  So  that  the  reaction  of  the  condenser  will  lead  the 
current  by  90°  and  the  component  of  the  impressed  E.M.F.  to 
overcome  this  reaction  will  lag  behind  the  current  by  90°. 

The  general  reaction  equation  then  of  the  alternating-current 
circuit  is 

W        T  di  4-  7?V  4-  Q 

EssL&  +  B*  +  C' 

and  as  i  =  -^j-  the  differential  equation  of  reaction  becomes 

dQ      Q 


from  which  the   value  of  current  may  be  obtained  for  any 


REACTIONS  IN  AN  A.  C.  CIRCUIT  27 

circuit  of  known  constants,  when  the  impressed  E.M.F.  and 
frequency  are  given. 

From  this  analysis  it  is  seen  that  the  impressed  E.M.F.,  treated 
as  a  vector,  will  have  three  components,  IR  in  phase  with  cur- 
rent, 2  7T/L7  leading  current  by  90°,  and  0  fri  lagging  90°  behind 

£i  7T/O 

the  current.  As  a  matter  of  fact,  the  total  reaction  of  neither  the 
inductance  nor  condenser  will  be  exactly  90°  out  of  phase  with 
the  current  because  of  losses  which  occur  in  them,  which  losses 
must  be  supplied  by  a  component  of  the  impressed  force  in  phase 
with  the  current.  But  whatever  the  relative  phases  of  the  react- 
ing forces,  they  must  always  add  up  as  vectors  to  give  the  neg- 
ative of  the  impressed  E.M.F.  plotted  as  a  vector. 

Connect  to  an  A.C.  supply  a  noninductive  resistance,  an  in- 
ductance and  a  condenser,  in  series  with  an  ammeter  and  the 
current  coil  of  wattmeter.  With  an  A.C.  voltmeter  measure  the 
impressed  E.M.F.  and  drop  in  potential  across  each  part  of  the 
circuit.  Also  read  the  power  consumption  in  the  whole  circuit 
and  in  each  part  and  read  current.  Keep  impressed  voltage  and 
frequency  constant  while  volts  and  watts  are  being  read;  take 
another  set  of  readings,  leaving  L,  R  and  C  the  same  but  with  / 
changed  and  see  how  the  different  reactions  change  with  a  change 
in  /.  Take  another  set  of  readings  after  having  adjusted  L,  R 
and  C  to  different  values. 

With  the  readings  obtained  construct  vector  diagrams  to  see 
whether  or  not  the  component  E.M.F. 's  add  to  give  the  impressed 
E.M.F.  in  both  magnitude  and  phase. 


EXPERIMENT   VIII. 
OHM'S  LAW  APPLIED  TO  THE  ALTERNATING-CURRENT  CIRCUIT. 

THE  quantity  by  which  the  impressed  A.C.  voltage  of  any 
circuit  must  be  divided  to  give  as  quotient  the  current  flowing 
in  the  circuit  is  called  the  impedance  of  the  circuit  and  generally 
designated  by  Z.  This  quantity  is  generally  expressed  in  ohms, 
although  it  really  involves  the  units  of  inductance  and  capacity 
as  well  as  the  ohm.  By  definition  of  Z  we  have,  /Z  =  E,  but 
in  the  preceding  experiment  it  was  shown  that  E  was  generally 
made  up  of  three  components  to  balance  the  three,  reactions  in 
the  general  A.C.  circuit.  Using  the  prefixed  letter  j  to  designate 
vectors  rotated  90°  in  a  clockwise  direction  and  —  j  to  indicate 
counter  clockwise  rotation,  the  equation  of  reactions  may  be 
written 

E  =  IR  +    _ 


From  this  it  may  be  seen  that  Z,  considered  from  the  stand- 
point of  reactions,  is  a  complex  quantity  and  has  direction,  but  as 
Z  is  ordinarily  used  as  a  simple  magnitude,  and  not  as  a  vector, 
it  is  necessary  to  find  the  magnitude  of  the  above  vector  and  call 
this  value  Z. 

As  the  two  components  of  Z  in  the  above  expression  are  at 
right  angles  to  one  another  it  is  evident  that  the  magnitude  of 
Z  is  given  by  the  expression 


Z=\l  K*  +  (2TT/L- 


-Y, 


27T/C/ 

and  so  for  the  A.C.  circuit  containing  resistance,  inductance  and 
capacity  in  series,  Ohm's  law  becomes 

E  E 


+  /WfL         l 

-     &  TTJlj   —  pr 


28 


LAW  FOR  CURRENT  IN  A.   C.   CIRCUIT  29 

and  the  phase  difference  of  /  and  E  is  given  by  equation 


_ 


, 
I  Jb 


When  dealing  with  circuits  having  two  or  more  branches  in 
parallel  it  becomes  convenient  to  introduce  some  new  terms. 
Corresponding  to  the  factors,  resistance,  react- 
ance  and  impedance,  we  shall  have  the  terms 
conductivity,  susceptance  and  admittance,  des- 
ignated  respectively  by  the  symbols  g,  b  and  Y. 
The  significance  and  use  of  these  terms  will  now 
be  taken  up. 

Referring  to  Fig.  12,  which  shows  two  in- 
ductances in  parallel,  it  is  evident  that  the  line 
current  /  is  equal  to  the  vector  sum  of  the 
branch  currents  ia  and  4.  If  the  potential  dif- 
erence  of  the  extremities  of  the  parallel  path  is  E,  it  is  seen  at 
once  that 

•  E 


and  that  it  is  made  up  of  two  components,  one  in  phase  and  one 
90°  out  of  phase  with  E. 

r  FV 

ia  (in  phase)  =  ia  cos  0  =  ia  -^-  =  y|  • 

ia  (90°  out  of  phase)  =  ia  sin  <£  =  -y~  - 
Collecting  these  expressions  we  have 

ia  (total)  =  %-  =  EYa, 
where  Ya  is  called  admittance  of  circuit  a; 

77T 

ia  (in  phase)  =  -=-%  =  Egai 


where  ga  is  called  conductance  of  circuit  a; 


ia  (90°  out  of  phase) 


7TT 


Eba, 


where  ba  is  called  susceptance  of  circuit  a. 

If  now  E  is  taken  as  having  a  magnitude  of  one  volt  the 
meanings  of  Y,  g  and  6,  become  apparent.     They  are  the  total 


30  ALTERNATING  CURRENTS 

current,  in-phase  current,  and  out-of-phase  current,  respectively, 
per  unit  E.M.F.  impressed  on  the  circuit. 

In  the  same  way  as  given  above  Yb,  gb  and  bb  are  obtained. 
'As  the  line  current  is  the  vector  sum  of  currents  through  the 
separate  paths,  and  it  is  easiest  to  treat  vector  quantities  by 
using  their  X  and  Y  components,  we  shall  have 

Line  current  (in  phase)  =  E(ga  +  gb)  =  EG. 
Line  current  (out  of  phase)  =E(ba  +  bb)  =EB. 

T> 

Lag  angle  of  line  current  = 


The  same  method  of  adding  conductances  and  susceptances  is 
employed  for  any  number  of  paths  in  parallel. 

If  these  parallel  circuits  are  in  turn  in  series  with  other  cir- 
cuits it  becomes  necessary  to  express  G  and  B  in  terms  of  re- 
sistance and  reactance. 

For  any  circuit  the  in-phase  current  is  given  by  the  expression 

T  /•       i       \       E  E  R 

I  (in  phase)  =  ^  cos  0  =  -^-  > 

T  /         f    i_      \      E  .  EX 

I  (out  of  phase)  =  ^  sm  <j>  =  -^  > 

/  (total)  =  |  =  EY. 

7?  JT 

From  this  is  obtained  G  =  ^  and  B  =  ^  where  R  and  X  repre- 

sent the  resistance  and  reactance  of  the  combined  parallel 
paths.  Hence,  to  reduce  the  two  parallel  paths  to  an  equivalent 
single  path  having  characteristics  R  and  X}  we  have: 


Y2  Y2 

IlL  _l_  1*-  r°     \     Tb 

Za2  """  Zb2  Z2  *  Zb2 


( r         r,  \2     / x         x  \2 

_li.  _|_  — 5.  1  _l_  f  Jl£.  4-  —  I 

In  a  similar  fashion 


LAW  FOR  CURRENT  IN  A.  C.  CIRCUIT 


31 


If  a  third  circuit  c  is  in  series  with  this  parallel  path  as  shown 

V  fi1 

in  Fig.  13,  the  current  7  =  -^  =     .  In  case 

Z      V(r. +  «)'+(*«+ X)' 

a  condenser  is  used  in  any  part  of  the  circuit  its  reactance,  xt 
must,  of  course,  be  taken  as  negative. 

The  solution  of  parallel  circuits  can  also  be  obtained  by  vector 
construction  and  generally  this  is  easier  than  the  analytical 
method  given  above.  The  solution  of  parallel  circuits  by  the 
use  of  graphical  methods  is,  in  fact,  so  much  easier  than  by  the 
analytical  method  that  the  latter  is  only  to  be  recommended  in 
the  simplest  cases. 

The  graphical  method  of  solution  is  based  on  the  fact  that  the 
two-branch  currents  may  be  added  as  vectors  to  give  the  line 
current  in  magnitude  and  phase;  no  matter  how  many  circuits 
join  at  some  branch  point,  the  vector  sum  of  all  currents  flowing 
away  from  the  point  must  equal  that  of  the  currents  flowing  to 
it.  If  the  currents  of  branches  a  and  6  (Fig.  12),  are  added 
vectorially,  as  in  Fig.  14,  the  line  current  is  given  by  the  vector 


o 


FIG.  13. 


FIG.  14. 


01,  both  in  magnitude  and  proper  phase  with  respect  to  the 
voltage  vector  OE.  If  now  the  currents  ia  and  ib  are  taken  of 
proper  magnitude  to  represent  the  respective  current  per  unit 
potential  difference  on  the  two  circuits,  then  01  is  the  line 
current  per  volt  impressed  on  the  circuit.  Now  the  reciprocal 
of  the  current  per  volt  01  will  give  the  impedance  of  the  com- 
bined paths  in  ohms. 

By  adding  vectorially  the  "  amperes  per  volt "  of  each  circuit 
connected  to  the  branch-point  considered,  the  "  line  current 
per  volt  "  is  obtained  and  the  reciprocal  of  this  quantity  is  the 


32 


ALTERNATING  CURRENTS 


impedance  in  ohms;  the  components  of  this  impedance  in  the 
F  and  X  axis  give  the  reactance  and  resistance  of  the  parallel 
circuit;  to  these  components  may  be  added  the  reactance  and 
resistance  of  whatever  circuit  is  in  series  with  the  parallel  path, 
as  c  (Fig.  13),  and  the  constants  for  the  complete  circuit  are  thus 
obtained.  t 

With  known  values  of  L,  R  and  C,  arrange  series,  series-parallel 
and  parallel  circuits  as  shown  in  Fig.  15.  With  suitable  volt- 
meter, ammeter  and  wattmeter  measure  the  impressed  E.M.F., 


the  line  current  and  power  used  in  the  combined  circuit.  From 
the  known  constants  of  the  different  parts  of  the  circuit  cal- 
culate what  current  should  flow  under  the  pressure  E  and  its 
phase  with  respect  to*  E,  for  each  of  the  three  cases,  using  one  of 
the  methods  just  described.  Compare  with  the  measured  values. 
In  this  test  it  is  well  to  use  an  air  core  inductance,  otherwise 
L  must  be  measured  for  each  value  of  the  current  used  in  the 
experiment. 


EXPERIMENT   IX. 

CIRCLE  DIAGRAM  FOR  CIRCUIT  CONTAINING  RESISTANCE  AND 

REACTANCE. 

JUST  as  it  is  possible  to  predict  the  characteristics  of  direct- 
current  machines  when  certain  constants  of  the  machine  are 
known,  so  can  the  behavior  of  alternating-current  machinery  be 
predetermined. 

For  such  machines  as  the  induction  motor,  series  motor  and 
transformer,  it  is  not  necessary  to  use  the  method  of  calculation 
to  predetermine  the  characteristics  of  the  machine,  but  they  may 
be  determined  graphically  from  the  "  circle  diagram."  As  this 
diagram  is  of  considerable  importance  in  experimental  work  it 
is  necessary  that  the  validity  of  the  construction  be  tested. 

The  fundamental  idea  involved  in  the  circle  diagram  is  this: 
If  any  alternating  E.M.F.  of  constant  magnitude  and  frequency  be 
applied  to  a  circuit  containing  resistance  and  reactance  in  series, 
either  one  of  which  is  varied  (the  other  remaining  constant), 
then  the  locus  of  the  current  flowing  in  the  circuit  will  be  a  circle. 
That  this  is  true  may  be  shown  mathematically  as  follows: 


where  X  designates  whatever  reactance  (inductance  or  capacity) 
the  circuit  may  contain  and  this  reactance  is  to  be  a  constant 
value  while  R  will  vary.  We  may  write 


=-cos 


X      VR*  +  X2      X 

E 

where  •==  is  a  constant  quantity  and  6  is  the  complement  of  the 

angle  between  current  and  E.M.F.    Such  an  equation  is,  of  course, 
that  of  a  circle  f  of  diameter  =  ^J,  expressed  in  polar  coordinates. 

If  then  R  is  varied  through  a  wide  range  of  values,  /  will  be 
given  by  the  series  of  chords,  drawn  from  one  end  of  the  diameter 

(of  magnitude  ^J,  to  the  circle  constructed  on  this  diameter  as 

in  Fig.  16. 

33 


34 


ALTERNATING  CURRENTS 


As  6  in  the  above  equation  is  the  complement  of  the  angle 
between  current  and  impressed  E.M.F.  the  proper  phase  relation 
between  I  and  E  will  be  given  by  letting  E  be  represented  by 
a  perpendicular  constructed  on  that  end  of  the  diameter  from 
which  the  chords  are  drawn. 

The  diameter  of  the  circle  will  be  the  maximum  possible  value 
of  the  current,  i.e.,  when  R  is  reduced  to  zero,  in  which  case  the 
phase  difference  of  E  and  I  becomes  90°.  This  is  as  it  should 


\  X  i 

FIG.  16. 

be,  because  no  power  can  be  used  in  a  circuit  containing  only 
reactance.  The  projections  of  the  vector  /  upon  the  diameter 
and  upon  the  vector  E  will  give  its  respective  wattless  and  power 
components. 

Connect  a  known  air-core  inductance  in  series  with  a  variable 
noninductive  resistance  which  can  be  changed  throughout  a 
wide  range.*  A  suitable  ammeter  and  wattmeter  must  be  used 
to  get  the  current  and  power  used  in  the  circuit  and  a  voltmeter 
to  read  the  impressed  voltage  and  see  that  it  remains  constant. 

Vary  R  through  as  wide  a  range  as  possible  (reducing  it  to 
as  near  zero  as  is  safe  with  the  apparatus  being  used),  reading 
amperes  and  watts  for  each  adjustment  of  R. 

Make  the  same  run  with  an  iron  core  inductance,  in  which 
case  L  will  generally  decrease  with  an  increase  of  current. 

Calculate  the  value  of   X  and  construct  a  semicircle  on  a 


Tjl 

diameter  of  the  magnitude  •= 

-A. 


Lay  off  E  (to  any  convenient 


scale)   perpendicular   to   this   diameter   at   its   left   extremity. 
From  the  same  end  of  the  diameter  lay  off  a  series  of  lines 

*  As  an  air-core  coil  does  not  have  much  inductance  it  will  probably  be  best 
to  use  an  alternating  current  of  the  highest  frequency  obtainable. 


CIRCLE  DIAGRAM  35 

making  with  E  angles  whose  cosines  are  calculated  from  the 
observations  of  watts  and  volt-amperes.  Upon  each  line  lay 
off  (to  the  same  scale  as  the  diameter  of  the  semicircle)  its 
respective  value  of  current  to  see  how  nearly  the  semicircle 
comes  to  being  the  locus  of  the  current  under  the  different  con- 
ditions, both  for  the  constant  L  and  variable  L. 

What  would  be  the  effect  upon  the  diagram  of  a  decreasing 
reactance  with  increasing  current?  Such  an  effect  actually 
exists  in  A.  C.  machinery;  the  demagnetizing  effect  of  the  rotor 
currents  of  an  induction  motor  act  to  produce  an  apparent 
decrease  in  the  inductance  as  the  load  is  increased. 


EXPERIMENT   X. 

FREE  AND  FORCED  VIBRATIONS;  RESONANCE  IN  A  CIRCUIT 
CONTAINING  RESISTANCE,  INDUCTANCE  AND  CAPACITY. 

ANY  system,  mechanical  or  electrical,  having  concentrated 
mass  and  an  elastic  restoring  force  proportional  to  the  displace- 
ment, will  be  capable  of  oscillating  continually  in  one  definite 
frequency,  executing  simple  harmonic  motion.  Translated  into 
electrical  terminology,  this  means  that  a  condenser  and  induct- 
ance, connected  in  series,  as  shown  in  Fig.  17,  the  condenser 
being  charged,  form  a  circuit  in  which,  when  the  switch  A  is 
closed,  current  will  surge  back  and  forth  with  a 
-  definite  frequency  and  will  be  a  sine  function 
of  the  time.  At  that  time  when  the  conden- 
ser is  fully  charged  there  is  no  current  in  the 


UJ'c 


FIG   17  circuit  and  all  of  the  energy  exists  as  potential 

energy  of  the  charged  condenser.  A  quarter 
of  a  cycle  later  the  condenser  is  discharged  and  so  possesses 
no  energy,  but  at  this  time  the  current,  and  hence  the  mag- 
netic field  of  the  inductance,  will  have  a  maximum  value,  and 
all  of  the  previous  potential  energy  of  the  condenser  will  be 
stored  as  kinetic  energy  of  the  magnetic  field.  If  there  is  no 
resistance  (effective)  in  the  circuit  (i.e.,  no  energy  is  dissipated 
either  as  heat  or  Hertzian  waves)  the  circuit  will  continually 
oscillate,  the  frequency  of  oscillation  being  fixed  by  the  con- 
stants of  the  circuit  as  will  now  be  shown  : 

If      Q  =  charge  on  condenser  before  switch  is  closed, 
i  =  current  in  circuit  at  any  time, 
L  =  coefficient  of  self-induction  of  the  coil, 
C  =  capacity  of  condenser, 

we  have,  as  soon  as  the  switch  is  closed  and  condenser  begins 
to  discharge,  i  =  -^  • 

As  the  sum  of  the  two  reactions  in  the  circuit  must  equal  the 
impressed  force,  and  this  is  zero, 

di      Q  t      d*Q     Q 

L^  +  c  =  0'or  L^+c  =  0- 

36 


FREE  AND  FORCED  VIBRATIONS  37 

The  solution  of  this  differential  is  obtained  by  assuming  Q  = 

A  cos  cot  and  so  -^  =  —  Au2  cos  cat. 
at2 

Substituting  these  values  in  the  original  equation 

A 

—  L  Aco2  cos  coZ  +  -^  cos  cot  =  0,  from  which 
C 


so  that,  as  2  irf  =  co,  we  have  /  =  =  -  7=  as  the  frequency  of 

*TT\/LC 

the  free  vibrations  in  such  a  circuit.  Another  solution  may 
be  obtained  by  putting  Q  =  B  sin  tat,  and,  therefore,  the  com- 
plete solution  becomes 

i  =  -j-f  =  -co  A  sin  cot  H-  coB  cos  co£. 
<U 

To  get  the  value  of  the  two  constants,  A  and  B,  we  will  assume 
that  when 

i  —  0  (i.e.,  before  switch  is  closed),  i  =  0, 
so  that 

0  =  0  +  co£  cos  at     or   B  =  0. 

If  at  time  •=  the  value  of  i  is  called  Im, 

Im  =  —  coA  or  A=  —  -> 

CO 

so  that  the  particular  solution  becomes 


Of  course  it  is  impossible  to  so  construct  a  circuit  that  no 
energy  is  dissipated  while  the  system  is 
oscillating.  It  is,  therefore,  necessary  to 
introduce  into  the  equation  of  reactions  a 
dissipative  term,  proportional  to  the  cur- 
rent. The  circuit,  then  depicted  by  Fig.  18, 
will  have  for  its  equation  of  reactions 

<PQ         dQ     Q 


The  solution  of  such  an  equation  requires  more  mathematics  than 
is  thought  well  to  introduce  in  a  text  intended  for  laboratory 


38  ALTERNATING  CURRENTS 

use,  but  the  solution  is  given: 


where  0  depends  upon  the  time  from  which  t  is  reckoned. 

This  solution  is  somewhat  different  from  the  previous  one. 
The  exponential  term  will  evidently  produce  damping,  so  that 
the  vibrations  gradually  die  out,  their  energy  being  dissipated 
from  the  system  in  the  form  of  heat  or  electric  waves.  The 
frequency  of  the  vibration  is  given  by  the  cosine  term  so  that 


. 

LC      4L2' 

which  is  a  slower  vibration  than  that  obtained  for  the  non- 
damped  circuit.  In  fact,  if  the  resistance  is  too  high,  no  oscilla- 
tion of  current  takes  place;  the  current  dies  out  without  ever 
reversing  its  direction. 

It  is  to  be  noticed  that  the  damping  and  difference  in  fre- 
quency from  the  first  case  are  not  affected  by  the  resistance  alone 
but  by  the  ratio  of  the  resistance  to  the  inductance.* 

The  general  case  is  now  considered.  A  circuit  as  shown  in 
Fig.  11  has  impressed  upon  it  an  alternating  E.M.F.  The  equa- 
tion of  reactions  becomes 


The  complete  solution  of  this  equation  gives  two  terms,  one 
having  a  frequency  equal  to  that  of  the  impressed  force  and  the 
other  having  a  frequency  equal  to  that  just  obtained  for  the 
free  vibration  of  such  a  system. 
The  complete  solution  is 


The  angle  e  depends  upon  the  constants  of  the  circuit  and  A 
depends  upon  the  instantaneous  value  of  the  impressed  force 
when  the  switch  is  closed  on  the  circuit.  Its  value,  however, 
will  never  be  greater  than 

E 


*  For  curves  showing  this  oscillatory  discharge  and  the  effect  of  resistance, 
see  Appendix,  Plates  1  and  2. 


FREE  AND  FORCED  VIBRATIONS 


39 


As  will  be  seen,  the  second  term  soon  becomes  negligible  be- 
cause of  its  damping  coefficient,  so  that  for  the  steady  state 
(which  is  the  condition  investigated  in  the  laboratory)  the 
solution  reduces  to  the  well-known  form: 

Tjt 

i  =  -^sin  (co/  —  e). 
£t 

It  is  interesting,  however,  to  investigate  the  significance  of 
the  second  member  of  the  solution.  In  the  steady  state,  the 
current  i  has  a  certain  fixed  amplitude  and  a  certain  angle  of 
displacement,  with  respect  to  the  impressed  force.  In  Fig.  19, 
the  full  line  marked  E  represents  the  impressed  force,  the  full 
line  marked  i  represents  the  steady  value  of  the  current  in  its 


FIG.  19. 


proper  phase  with  respect  to  E.  It  is  quite  evident  that  at  the 
time  of  closing  the  switch  which  connects  the  source  of  E.M.F. 
to  the  circuit,  the  current  must  have  zero  value.  If  the  switch 
is  closed  at  time  T\,  the  current  i  will  have  its  proper  zero  value 
and  proper  phase  with  respect  to  E,  and  if  the  condenser  should 
happen  to  be  charged  to  the  potential  difference  that  normally 
exists  across  it  at  that  part  of  the  cycle  indicated  by  T^  then  the 
transient  or  exponential  term  will  reduce  to  zero;  it  never  exists. 
In  general,  the  switch  will  not  be  closed  at  this  time  but  at  some 
such  time  as  marked  T2.  The  steady  current  should  have  at  this 
time  a  value  equal  to  A.  But,  as  before  mentioned,  the  actual 
current  must  be  zero  at  the  time  of  closing  the  switch  and  so 
something  must  happen  to  bring  the  actual  current  to  the  proper 
magnitude  and  phase  to  fit  the  steady  state.  This  is  the  function 
of  the  transient  term.  The  actual  current  which  flows  when  the 
switch  is  closed  at  time  T2  is  shown  by  /,  the  transient  damped 
term  by  /',  while  i  represents  the  steady  current.  The  damped 
current  /',  which  represents  the  free  vibration  of  the  system,  will 


40  ALTERNATING  CURRENTS 

have  a  period  not  depending  upon  that  of  the  impressed  force 
but  upon  the  constants  of  the  circuit.  Its  amplitude  depends 
upon  the  constants  of  the  circuit  and  the  time  of  closing  the 
switch.  It  will  always  so  act  that  the  actual  current  7  is  soon 
brought  into  coincidence  with  the  steady  current  i.  If  the 
damping  of  the  circuit  is  high  this  may  take  place  within  a  few 
alternations  (or  even  in  one)  but  if  the  resistance  of  the  circuit 
is  low  the  exponential  term  may  last  for  perhaps  a  hundred 
alternations.* 

It  is  the  purpose  of  this  experiment  to  investigate  the  possible 
values  of  this  steady  current,  so  the  transient  term  is  neglected. 
We  have 

E 

:sin  (cot  —  c,) 


or,  in  effective  values, 

7  =  * 


When  the  impressed  frequency  =  s  --  7=  ,  coL  =  —^  ,  in  which 


Tjl 

case  the  equation  reduces  to  I  =  -=  ,  and  the  circuit  is  said  to  be 

K 

in  resonance. 

As  many  A.  C.  circuits  have  very  low  resistances,  the  value  of 
the  current  under  such  conditions  may  become  very  high.  As 
the  potential  difference  across  the  terminals  of  the  inductance 
and  capacity  depends  directly  upon  7,  it  may  become  of  such 
high  value  that  the  insulation  of  the  line  is  broken  down.  In 
a  circuit  having  inductance  and  condenser  in  series,  the  drop  of 
potential  across  either  the  inductance  or  condenser  may  be  many 
times  larger  than  the  E.M.F.  impressed  upon  the  circuit.  In 
practice  such  a  condition  might  arise  from  the  capacity  of  a  cable 
of  low  resistance  and  inductance  of  a  transformer  coil. 

Transmission  line  break-downs  are  often  attributed  to  reso- 
nant rise  in  potential  due  to  some  high  current  set  up  in  it  by 
an  arcing  short  circuit,  higher  harmonics  in  the  alternators, 
etc.  As  a  transmission  line  consists  of  distributed  capacity, 
resistance,  and  inductance,  it  is  doubtful  if  real  resonance  can 
occur  unless  the  resistance  and  inductance  of  the  line  are  low 
*  For  curves  illustrating  this  point  see  Plates  3  and  4. 


FREE  AND  FORCED  VIBRATIONS 


41 


and  some  such  inductance  as  a  transformer  or  an  alternator  is 
attached  to  it.  In  such  case  the  line  simply  acts  as  a  condenser. 
A  kind  of  resonance  may  occur  on  a  transmission  line  if  the 
length  of  the  line  is  such  that  reflected  waves  return  in  the  right 
phase  to  be  reinforced  by  the  impressed  E.M.F.  With  ordinary 
frequencies  the  longest  transmission  line  is  not  long  enough  for 
such  an  effect,  so  that  this  kind  of  building  up  of  the  voltage  does 
not  occur  unless  the  alternator  has  a  marked  higher  harmonic 
of  such  frequency  that  the  above  condition  is  fulfilled.  In  such 
a  case  the  upper  frequency  current  will  be  much  accentuated 
and  may  become  of  great  enough  value  to  cause  failures  of  in- 
sulators, etc. 

When  the  inductance  and  capacity  are  connected  in  series  as 
in  the  circuit  just  analyzed,  the  current  in  the  line  may  reach 


120  3 
80  2 
40  1 


V/ 


0          80         100 
Frequency 

FIG.  20. 


120 


very  high  values  and  so  the  drop  of  potential  across  either  part 
of  the  circuit  may  be  correspondingly  high. 

Another  type  of  resonant  circuit  is  of  interest;  the  inductance 
and  condenser  are  connected  in  parallel  and  then  to  the  supply 
line  as  shown  in  Fig.  21.  In  this  case  the  impedance  of  the  joint 
path  may  be  very  high  so  that  only  a  small  current  flows  in  the 
line.  But  the  inductance  and  condenser  are  in  series  on  a  local 
circuit  and  if  the  line  frequency  is  such  that  resonance  would 
occur  for  the  series  connection  then  resonance  will  nearly  occur  for 
this  parallel  circuit.  The  voltage  drop  across  either  the  condenser 
or  inductance  cannot  be  greater  than  the  voltage  of  the  line,  the 
resonant  condition  is  shown  by  the  relative  current  values  in  the 
line  and  local  circuit.  When  resonance  occurs  the  ammeter  A 


42  ALTERNATING  CURRENTS 

may  read  many  times  greater  than  the  line  ammeter.  It  is  to 
be  noted  that  in  these  circuits  tuned  for  resonance  the  resistance 
in  the  oscillating  circuit  is  the  only  thing  that  limits  the  value 
to  which  the  current  will  build  up.  In  the  parallel  circuit  the 
value  of  the  critical  frequency  (that  which  gives  minimum  watt- 
less current  in  the  line)  is  affected  by  the  resistance  in  the  oscil- 
lating circuit.  When  the  charging  current  of  the  condenser  is  just 
equal  to  that  of  the  inductance,  we  have 


which  gives  as  the  resonant  period  of  the  system 


LC(CRC2-L) 

Connect  in  series  an  inductance,  condenser  and  resistance  and 
apply  to  the  terminals  of  the  circuit  a  source  of  variable  fre- 
quency E.M.F.  Calculate  suitable  values  for  L  and  C  (from  the 
formula  given  in  the  first  part  of  this  experiment)  so  that  for  a 
value  of  frequency  at  about  the  middle  value  of  the  obtainable 
range 


Connect  an  ammeter  in  series  with  the  circuit  and  across  the 
condenser  connect  a  static  voltmeter.  Use  an  ordinary  volt- 
meter to  keep  the  impressed  voltage  constant  and  to  measure 
the  voltage  across  the  inductance.  With  a  fairly  high  value  of 
resistance  in  the  circuit  vary  the  frequency  (in  about  10  steps) 
through  the  range  obtainable,  reading  volts 
across  the  condenser  and  inductance  and  the 
current.  Take  a  similar  set  of  readings  using  a 
resistance  of  about  one-fourth  the  previous  value 
(providing  the  apparatus  used  will  stand  such  a 
low  resistance). 

Make  similar  runs  for  the  parallel  circuit  de- 
picted in  Fig.  21. 
FlQ  21  Pl°t  curves  between  values  of  current,  induc- 

tance drop  and  condenser  drop  against  frequency 
as  abscissae  for  the  series  connection  test  and  for  the  parallel 
connection,  curves  of  line  current  and  local  circuit  current,  using 


FREE  AND  FORCED  VIBRATIONS  43 

the  same  scale  of  abscissae  as  for  the  first  set  of  curves.  The 
curves  should  have  about  the  same  appearance  as  those  given 
in  Fig.  20,  which  represent  conditions  of  current  and  voltage 
for  the  series  circuit  with  low  value  of  R.  For  larger  values  of 
R  the  resonance  is  not  so  marked. 


EXPERIMENT   XI. 

MAGNETIZATION  CURVE  (NO-LOAD  SATURATION  CURVE)  OF  AN 

ALTERNATOR  AND  EXTERNAL  CHARACTERISTIC  ON 

VARIOUS  POWER  FACTORS. 

THE  magnetization  curve,  showing  the  relation  between  volt- 
age generated  by  the  armature  and  field  current,  is  useful  to  the 
designer,  showing  to  what  extent  the  field  of  the  machine  is 
saturated  (thus  giving  an  idea  as  to  whether  too  much  iron  or 
not  enough  has  been  used  on  the  design  of  the  magnetic  circuit), 
and  is  also  valuable  as  being  one  of  the  curves  from  which  the 
external  characteristic  of  the  machine  is  to  be  predetermined. 

For  constant  speed  (under  which  condition  all  of  the  above 
curves  are  to  be  taken)  the  generated  voltage  is  directly  pro- 
portional to  the  flux  threading  the  armature.  This  being  the 
case,  a  curve  showing  the  relation  between  armature  E.M.F.  and 
field  current  really  shows  the  relation  between  the  field  current 
and  flux  through  armature,  i.e.,  gives  the  saturation  curve  of  the 
magnetic  circuit.  Not  all  of  the  flux  generated  by  the  field 
current  cuts  the  armature  conductors.  A  part  of  it  leaks  across 
from  one  pole  to  the  adjacent  pole  without  going  through  the 
armature  core.  This  is  called  the  leakage  flux,  and  the  factor 
by  which  the  total  flux  must  be  divided  to  give  the  flux  through 
the  armature  core  is  called  the  leakage  factor.  If  this  leakage 
factor  remains  constant  with  varying  field  excitation  the  arma- 
ture voltage  and  field  current  actually  do  give  the  proper  relation 
between  field  current  and  generated  flux;  but  if  the  leakage 
factor  increases  with  increase  of  voltage  (likely  to  occur  if  the 
armature  is  worked  at  high  flux  density  at  normal  voltage)  then 
with  the  higher  value  of  field  current  the  saturation  curve  will 
give  a  value  of  flux  which  is  increasingly  lower  than  the  actual 
flux  through  the  field  poles  and  frame.  It  will,  however,  give 
correctly  the  relation  between  field  current  and  flux  in  armature 
core. 

For  efficient  design,  so  far  as  the  iron  used  in  construction  is 
concerned,  the  normal  operating  field  current  should  give  a 

44 


THE  ALTERNATING  CURRENT  GENERATOR      45 

point  on  the  saturation  curve  somewhat  above  the  knee,  but 
such  a  density  in  the  armature  will  give  iron  losses  too  large  and 
necessitates  too  high  a  value  of  field  current  for  proper  efficiency 
of  operation.  A  suitable  density  is  one  which  operates  the 
magnetic  circuit  as  a  whole  slightly  below  the  knee  of  the  mag- 
netization curve. 

Excite  the  alternator  field  by  potentiometer  connection  to  the 
laboratory  D.C.  power.  With  rated  speed  determine  value  of 
field  current  to  give  different  armature  voltages  (the  machine 
being  unloaded),  from  no  excitation  to  such  as  will  give  25  per 
cent  over  normal  voltage.  Read  meters  at  about  10  points, 
taking  the  readings  closer  together  where  the  saturation  curve 
begins  to  bend  over.  Obtain  the  curve  for  increasing  values  of 
field  current  only  as  this  is  the  curve  to  be  used  in  predeter- 
mination of  regulation.  Observe  usual  precaution  in  varying 
field  current. 

In  obtaining  the  first  external  characteristic  (as  </>  =  1)  use  a 
lamp  bank  or  water  rheostat  for  load,  and,  with  alternator 
running  at  rated  speed,  adjust  the  field  current  until  rated  volt- 
age is  obtained  with  full-load  current  flowing  through  armature; 
read  terminal  volts,  load  current,  and  field  current.  Keep  field 
current  at  this  value  throughout  test.  Decrease  load  until 
about  f  rated  current  is  flowing  and  read  armature  current  and 
terminal  volts.  Take  similar  readings  for  J,  J,  0  and  lj  of  full- 
load  current. 

For  power  factors  other  than  one,  a  synchronous  motor, 
driving  a  loaded  D.C.  generator,  is  to  be  used  as  a  load.  De- 
pending upon  the  value  of  field  current  supplied,  a  synchronous 
motor  will  draw  either  a  leading  or  lagging  current  from  its 
supply  line.  After  the  synchronous  motor  has  been  connected 
to  the  alternator,  adjust  the  motor  load  until  nearly  full-load 
current  is  flowing  from  alternator.  Then  decrease  the  field 
current  of  the  synchronous  motor  and  adjust  its  load  and  also 
the  field  current  of  the  alternator  until  the  terminal  voltage  of 
the  alternator  is  at  its  rated  value,  it  is  carrying  full-load  current 
and  its  actual  load  is  0.8  its  apparent  load  (i.e.,  cos  0  =  0.8, 
lagging). 

Read  terminal  volts,  load  current,  field  current  and  watts  of 
alternator.  Keep  alternator  field  current  constant  at  this  value. 
Decrease  the  load  on  synchronous  motor  until  about  f  full- 
load  current  is  flowing  from  alternator,  readjust  field  current 


46  ALTERNATING   CURRENTS 

of  synchronous  motor  until  cos  <f>  =  0.8  and  take  same  set  of 
readings  as  before  obtained  for  full-load  current.  Take  similar 
readings  for  J,  J  and  1J  loads.  (Zero  load  cannot  be  obtained 
when  synchronous  motor  is  used  for  load.)  With  full-load  cur- 
rent on  alternator  and  cos  <£  =  0.8  (leading  current),  obtained 
by  over-exciting  the  synchronous  motor  field,  obtain  the  exter- 
nal characteristic  of  the  alternator  for  a  leading  current,  using 
about  the  same  values  of  load  current  as  before  and  maintaining 
a  leading  load  current  of  power  factor  =  0.8.  Then  with  the  syn- 
chronous motor  disconnected,  obtain  reading  of  terminal  volts  for 
zero  load  points  on  the  two  load  curves  of  power  factor  =  .8. 

If  a  power-factor  meter  of  suitable  capacity  is  available  the 
adjustment  of  the  power  factor  of  the  load  may  be  accomplished 
by  readings  of  this  meter  instead  of  by  computing  the  ratio  of 
actual  watts  to  apparent  watts  as  described  above. 

Calculate  the  regulation  of  the  alternator  for  the  three  dif- 
ferent loads  used. 


EXPERIMENT   XII. 

FULL-LOAD  SATURATION  CURVE,  SHORT-CIRCUIT  CURRENT  (SYN- 
CHRONOUS-IMPEDANCE CURVE)  AND  ARMATURE 
CHARACTERISTIC. 

THE  full-load  saturation  curve  of  an  alternator  shows  the 
relation  between  the  field  current  and  terminal  voltage  of 
machine  when  full-load  current  is  flowing  in  the  armature  circuit. 
It  will  be  very  similar  in  form  to  the  no-load  saturation  curve 
but  will  of  course  be  lower  than  this  curve,  if  plotted  on  the  same 
curve  sheet.  The  no-load  saturation  curve  gives  the  total  E.M.F. 
generated  in  the  armature  for  different  values  of  field  current 
while  the  full-load  saturation  curve  records,  for  corresponding 
values  of  field  current,  the  vector  difference  between  the  gener- 
ated E.M.F.  and  the  impedance  drop  in  the  armature  due  to  full- 
load  current.  If  the  impedance  of  the  armature  remained 
constant  for  all  values  of  field  current  then  the  actual  difference 


R  E  E' 

OZ,  OG,  OG\  —Generated  volts 
O,  OE,  OEj — Terminal  volts 

FIG.  22. 

between  the  no-load  voltage  and  the  terminal  voltage  of  the 
generator  while  it  is  carrying  full-load  current  (for  the  same  value 
of  field  current)  would  grow  less  with  increasing  values  of  the 
exciting  current.  This  is  readily  seen  from  the  vector  diagram 
given  in  Fig.  22.  At  that  value  of  field  current  which  gives  a 
terminal  voltage  equal  to  zero  on  the  full-load  saturation  curve 
the  no-load  saturation  curve  gives  a  voltage  nearly  equal  to  the 
impedance  drop  in  the  armature,  so  that  the  difference  between 
the  two  is  practically  the  full-load  impedance  drop  in  the 

47 


48  ALTERNATING  CURRENTS 

armature;  but  at  higher  values  of  excitation  the  vector  subtrac- 
tion of  the  impedance  drop  has  much  less  effect  on  the  terminal 
voltage.  The  above  vector  diagram  is  constructed  on  the  as- 
sumption that  the  load  on  the  alternator,  when  the  full-load 
saturation  curve  is  taken,  is  noninductive,  the  condition  under 
which  this  test  is  to  be  run.  If  the  load  were  inductive  the  above 
remarks  about  the  difference  between  no-load  and  full-load 
saturation  curves  would  be  true  only  when  the  phase  angle  of 
the  load  is  less  than  the  phase  angle  of  the  armature  circuit. 
If  the  external  circuit  has  the  same  power  factor  as  the  armature 
circuit  then  the  difference  between  the  two  curves  would  be  the 
same  for  all  values  of  field  current. 

As  a  matter  of  fact  the  armature  impedance  decreases  slightly 
from  no  excitation  to  full  excitation  and  more  quickly  for  higher 
values  of  flux  density.  The  reason  for  this  lies  in  the  fact  that 
the  armature  teeth  form  a  large  part  of  the  path  of  the  leakage 
flux  to  which  the  armature  reactance  is  due.  The  teeth  carry 
the  normal  flux  of  the  machine  as  well  as  the  leakage  lines  and 
as  soon  as  the  sum  of  these  two  fluxes  becomes  so  large  that  the 
teeth  are  more  or  less  saturated  then  the  reluctance  of  the  path 
of  the  leakage  lines  increases  and  so  the  armature  reactance 
diminishes. 

The  short-circuit-current  curve  gives  the  relation  between  the 
current  in  the  armature  when  short  circuited  and  the  field  cur- 
rent. It  is  obtained  by  short  circuiting  the  armature  through 
an  ammeter  (of  range  about  100  per  cent  overload  for  the  alter- 
nator) and  gradually  increasing  the  field  current  until  the  safe 
carrying  capacity  of  the  armature  is  reached,  a  series  of  readings 
being  taken  of  armature  current  and  field  current.  Under  these 
conditions  it  is  quite  evident  that  all  of  the  voltage  generated 
in  the  armature  is  used  up  in  overcoming  the  armature  impedance 
drop.  The  impedance  of  the  armature  will  vary  with  the  fre- 
quency; as  the  data  from  which  this  curve  is  plotted  are  taken 
as  normal  frequency  of  the  machine,  it  is  called  the  synchronous- 
impedance  curve. 

The  impedance  of  the  armature  is  generally  calculated  from 
the  data  of  this  curve.  For  any  value  of  the  short-circuit 
armature  current  the  field  current  is  obtained;  the  generated 
voltage  due  to  this  field  current  is  taken  from  the  magnetization 
curve;  this  generated  voltage  divided  by  the  short-circuit  current 
is  called  the  synchronous  armature  impedance. 


THE  ALTERNATING  CURRENT  GENERATOR      49 

The  value  of  impedance  so  obtained  is  far  from  being  the  true 
armature  impedance  for  several  reasons.  Even  at  the  highest 
value  of  armature  current  permissible  in  this  test,  the  armature 
teeth  are  generally  far  from  being  saturated  because  the  field 
current  has  a  low  value.  Also  in  discussing  the  reactions  which 
occur  in  an  alternator  it  will  be  shown  that  a  lagging  current 
demagnetizes  the  -field,  the  demagnetization  being  the  greater, 
the  larger  the  lag  angle.  Now  the  armature  voltage  on  short 
circuit  is  nearly  all  used  up  in  overcoming  the  reactance  drop,  as 
the  reactance  is  much  the  larger  part  of  the  armature  impedance. 
Hence  on  short  circuit  the  current  lags  almost  90°  behind  the 
generated  E.M.F.  and  so  will  demagnetize  the  field  to  a  consider- 
able extent.  The  E.M.F.  taken  from  the  magnetization  curve 
will  be  much  larger  than  the  E.M.F.  actually  generated  in  the 
armature  when  short-circuit  current  is  flowing  and  so  the  arma- 
ture impedance  obtained  by  this  method  will  be  too  large. 

Another  method  of  getting  the  synchronous  impedance  is  to  run 
the  alternator  as  a  synchronous  motor  and  over-excite  its  field 
so  that  the  current  leads  the  impressed  E.M.F.  by  nearly  90°.. 
Under  such  conditions  the  impressed  voltage  and  induced  voltage 
are  nearly  180°  apart  and  the  IZ  drop  is  obtained  by  subtracting 
the  terminal  volts  from  the  induced  voltage.  The  induced 
voltage  is  obtained  from  the  magnetization  curve  of  the  machine 
for  that  value  of  field  current  which  is  required  to  bring  about 
the  above-stated  relation  between  impressed  and  induced  volt- 
age. In  this  method  also  no  account  is  taken  of  the  fact  that 
the  field  is  demagnetized  by  the  armature  current  and  as  the  field 
is  super-excited  the  teeth  are  more  saturated  than  under  normal 
conditions.  As  the  first  effect  would  give  too  high  a  value  for 
Z  and  the  second  too  low  a  value  this  method  probably  gives 
results  as  nearly  accurate  as  any  other  method  which  might  be 
employed. 

Also  the  value  of  Z  measured  will  vary  according  to  the  space 
position  of  the  armature  in  respect  to  the  phase  of  the  current. 
If  a  coil  side  happens  to  be  under  a  pole  face  at  the  time  the 
current  through  it  is  a  maximum,  different  value  of  Z  will  be 
obtained  than  if  the  coil  side  was  in  the  open  space  between 
adjacent  poles  when  the  current  was  a  maximum. 

From  the  foregoing  it  is  evident  that  the  synchronous  im- 
pedance of  an  alternator  is  a  somewhat  indefinite  quantity  and 
that  its  measurement  involves  some  rather  violent  assumptions. 


50  ALTERNATING  CURRENTS 

Many  writers  treat  it  as  a  fictitious  quantity  which  may  'be 
used  to  predict  the  alternator  regulation,  and  this  is  probably 
the  most  logical  way  of  treating  it. 

The  armature  characteristic  shows  the  relation  between  field 
current  and  load  current,  keeping  the  terminal  voltage  constant 
at  its  rated  value  for  all  loads.  If  the  magnetic  circuit  is  not 
near  saturation  this  curve  will  be  a  straight  line,  otherwise  (as 
e.g.,  when  the  no-load  voltage  is  somewhere  near  the  knee  of  the 
magnetization  curve)  the  curve  will  be  concave  upward,  it 
taking  a  greater  increment  of  field  current  for  a  given  increase 
of  load  current  near  full  load  than  near  no  load.  The  curve 
gives  the  engineer  an  idea  of  what  duty  the  exciters  will  have, 
how  much  they  should  be  compounded,  etc. 

To  get  the  full-load  saturation  curve  a  noninductive  resistance, 
which  may  be  varied  from  zero  to  a  value  equal  to  the  rated 
terminal  voltage  of  the  machine  divided  by  full-load  current,  is 
connected  across  the  armature  terminals  in  series  with  an  am- 
meter; this  resistance  must  be  capable  of  carrying  full-load 


FIG.  23 

current  of  the  alternator  throughout  its  range.  A  suitable  volt- 
meter is  connected  across  the  armature  terminals  as  shown  in 
Fig.  23.  The  field  is  excited  by  potentiometer  connection 
because  the  current  is  to  be  varied  throughout  a  wide  range. 
The  same  set-up  of  apparatus  is  to  be  used  for  all  three  curves, 
and  speed  is  to  be  kept  at  rated  value  for  all  three  tests.  For 
the  full-load  saturation  cut  out  all  of  the  resistances  in  the  arma- 
ture circuit  and  gradually  increase  the  field  current  from  zero 
until  full-load  current  is  flowing  through  the  armature.  Read 
armature  current,  field  current  and  terminal  voltage  (which  will 
be  zero  for  this  adjustment  of  load).  Put  some  resistance  in 
the  armature  circuit  and  adjust  this  resistance  and  field  current 
until  rated  current  is  flowing  m  the  armature  and  the  voltage  at 
terminals  is  about  J  of  the  rated  value  for  the  machine.  Take 


THE  ALTERNATING  CURRENT  GENERATOR      51 

same  readings  as  before.  Again  increase  load  resistance  and 
field  current  to  give  about  J  of  rated  voltage  with  full-load 
current  flowing.  Continue  this  until  about  25  per  cent  above 
rated  voltage  of  machine  is  reached. 

For  the  short-circuit-current  test  reduce  the  field  current  to 
zero  and  then  the  load  resistance  to  zero.  Leaving  the  load 
resistance  short  circuited,  increase  field  current  until  the  short- 
circuit  current  is  25  per  cent  full-load  value.  Read  field  current 
and  armature  current.  Increase  field  to  give  50  per  cent  rated 
current  in  armature  and  again  take  readings.  Continue  until 
armature  current  is  150  per  cent  full-load  value.  The  resistance 
of  the  external  circuit  must  be  reduced  as  low  as  possible  for  this 
test,  large  cables  should  be  used,  all  connections  well  made,  etc. 
Why? 

To  get  the  armature  characteristic,  adjust  the  field  current  to 
give  rated  voltage  at  no  load;  read  value  of  field  current  and 
voltage  of  armature.  Put  on  about  J  load,  using  noninductive 
resistance  for  load,  readjust  field  current  (increasing  its  value 
only;  if  the  proper  value  is  exceeded,  decrease  the  current  to  a 
small  value  and  bring  back  to  the  desired  adjustment)  to  give 
rated  terminal  voltage,  and  read  field  current,  load  current,  and 
terminal  voltage.  Continue  to  25  per  cent  overload  for  alter- 
nator. Make  a  similar  run  using  an  inductive  load  of  cos  0  =  0.8. 

Plot  all  curves  on  the  same  sheet,  using  for  the  first  curve 
terminal  volts  as  ordinates  and  field  current  as  abscissae;  for 
the  second  curve,  volts  on  open  circuit  (to  be  obtained  from 
magnetization  curve  and  field  current  read  in  this  test)  as 
ordinates  and  armature  current  as  abscissae;  for  the  armature 
characteristic  use  load  current  as  abscissa  and  field  current  as 
ordinates. 

Could  the  armature  characteristic  for  .load  of  cos  0  =  1, 
obtained  in  this  test  be  used  to  determine  the  capacity  of  the 
exciter  for  the  alternator,  if  the  alternator  was  to  be  used  for 
ordinary  commercial  loads,  such  as  a  central-station  load? 

Specify  a  suitable  rheostat  for  the  alternator  field,  assuming 

Note.  —  It  has  been  stated  above  that  practically  all  of  the  armature  im- 
pedance consists  of  its  reactance  component.  A  convenient  method  of  test- 
ing this  fact  is  to  short  circuit  the  armature  with  a  low  value  of  excitation  on 
the  field.  Now  vary  the  speed  of  the  alternator  through  a  wide  range  (do  not 
run  at  more  than  125  per  cent  of  rated  speed),  keeping  the  excitation  constant, 
and  read  the  current  circulating  in  the  armature.  Account  for  observed  facts. 


52  ALTERNATING  CURRENTS 

the  field  is  to  be  supplied  from  a  D.C.  line  of  proper  voltage  for 
the  given  field  and  the  load  on  the  alternator  is  to  be  a  mixed 
one,  induction  motors,  lights,  etc. 

If  the  armature  impedance  remained  constant  what  would  be 
the  short-circuit  current  of  the  alternator  with  normal  excitation? 


EXPERIMENT   XIII. 

METHODS  FOR  PREDETERMINING  THE  EXTERNAL  CHARACTER- 
ISTICS OF  AN  ALTERNATOR. 

IN  the  previous  tests  upon  the  alternator  it  has  been  shown 
that  the  drop  in  terminal  voltage  with  increase  of  load  is  due  to 
two  effects,  first,  the  actual  impedance  drop  caused  by  the 
resistance  and  reactance  of  the  armature;  second,  the  effect 
of  the  armature  current  upon  the  field  strength.  The  drop  in 
terminal  voltage  can,  therefore,  be  completely  accounted  for  by 
considering  first  the  voltage  drop  due  to  impedance  and  secondly 
the  change  in  magneto-motive  force  due  to  the  armature  ampere 
turns  reacting  upon  the  main  field.  If,  however,  the  whole 
effect  is  treated  either  from  the  standpoint  of  E.M.F.  or  of 
M.M.F.  the  external  characteristic  may  be  predicted  with  suffi- 
cient accuracy  for  loads  of  P.F.  =  1,  under  which  condition  of  load 
the  armature  reaction  tending  to  change  the  field  strength  of  the 
alternator,  is  negligibly  small.  Where  it  is  important  that  the 
effect  be  calculated  more  closely  than  either  the  E.M.F.  method 
or  M.M.F.  method  permits,  then  a  more  complicated  method 
of  treatment  is  adopted  in  which  each  of  the  components  of  the 
fictitious  "  synchronous  impedance  "  is  dealt  with  separately. 

The  value  of  these  methods  of  predicting  the  regulation  of  an 
alternator  is  realized  when  it  is  remembered  that  the  regulation 
of  a  machine  is  one  of  its  most  important  characteristics  and  that 
it  is  practically  impossible  to  actually  load  the  larger  machines 
with  a  load  of  any  desired  power  factor.  Also  the  energy  used 
makes  the  actual  loading  of  the  machine  an  expensive  method. 

The  E.M.F.  method  of  predicting  the  regulation  gives  results 
which  are  somewhat  worse  than  the  machine  will  actually  give 
and  so  is  termed  the  pessimistic  method,  while  the  M.M.F. 
method  gives  too  close  a  regulation  and  so  is  called  optimistic. 
For  either  method  the  two  curves  required  are  the  short-circuit- 
current  curve  and  the  magnetization  curve. 

The  E.M.F.  method  will  first  be  described  as  it  is  the  most 
important;  it  illustrates  the  application  of  the  vector  diagram  for 

53 


54 


ALTERNATING  CURRENTS 


solving  problems  in  A.C.  work  and  exactly  the  same  method 
may  be  used  for  predicting  the  regulation  of  transformers,  trans- 
mission lines,  etc.,  as  is  used  here  for  the  alternator.  The 
fictitious  "  synchronous-impedance  drop  "  is  considered  as 
though  made  up  of  two  components,  one  in  phase  with  the  current 
and  one  lagging  90°  behind  it.  The  synchronous  impedance  is 
calculated  as  given  in  Experiment  12.  The  ohmic  resistance  of 
the  armature  being  known  by  measurement  with  direct  cur- 
rent, the  synchronous  reactance  is  calculated  by  the  formula, 
X  =  VZ2  -R2. 

Then  to  predict  the  regulation  for  a  load  of  power  factor  =  1, 
the  operation  is  as  shown  in  Fig.  24.  The  phase  of  the  load 
current  is  assumed  as  01  and  the  rated  full-load  voltage,  OEt,  is 
laid  off  along  01  to  a  suitable  scale.  The  full-load  resistance 
drop  is  laid  off  to  the  same  scale  and  is  shown  in  Fig.  24  by  OR. 
With  a  radius  equal  to  the  value  of  the  full-load  synchronous 


FIG.  24. 

impedance  in  volts,  a  circular  arc  is  drawn  as  indicated  by  the 
dotted  line.  From  R  a  line  is  erected  perpendicular  to  01. 
The  origin  0  is  joined  by  a  straight  line  to  the  intersection  of 
this  perpendicular  and  the  circular  arc,  and  this  line  OZ,  repre- 
sents the  full-load  impedance  voltage  in  its  proper  phase  with 
respect  to  the  load  current.  The  vector  addition  of  OEt  and 
OZ  gives  the  vector  OB  which  represents  the  generated  voltage 
at  full  load  and  which,  in  this  prediction  method,  is  assumed  to 
remain  constant  for  all  loads.  With  0  as  center  and  OB  as 
radius,  a  circular  arc  is  constructed  as  shown,  and  this  curve 
is  the  locus  of  the  generated  voltage  as  the  load  varies.  To 

OZ 
predict    the    terminal   voltage    at    half  load,   take   OZ'  =  — ' 

ft 

(because  at  half  load  the  armature-impedance  drop  is  half  as 
great  as  it  is  for  full  load),  construct  Z'B'  parallel  to  07,  draw 
B'Et  parallel  to  OZ'  and  the  half-load  terminal  voltage  is  given 


THE  ALTERNATING  CURRENT  GENERATOR 


55 


by  the  vector  OEt'.  At  no  load  the  terminal  voltage  is  OEt"; 
for  any  other  load  the  same  construction  is  carried  out,  using  an 
impedance  drop  proportional  to  the  load  assumed.  The  charac- 
teristic constructed  from  points  obtained  from  the  vector  diagram 
will  lie  above  that  actually  measured  in  Experiment  11. 

The  regulation  for  cos  0  =  1,  is  given  by  the  quotient 
(OB  —  OEt)  -r-  OEt.  The  vector  construction  makes  plain  the 
process  of  predicting  the  regulation  but  will  not  generally  be 
accurate  enough,  so  that  the  problem  is  solved  analytically.  By 
reference  to  the  vector  diagram,  it  is  seen  that 


Regulation  =  (V(OEt 


-  OEt)  +  OEt. 


If  it  is  desired  to  predict  the  regulation  for  a  load  having 
power  factor  other  than  one  the  method  of  procedure  is  indicated 
by  Fig.  25.  Here  a  load  of  cos  0  =  0.866  (<j>  =  30°)  is  assumed. 


FIG.  25. 

As  before,  the  phase  of  the  load  current  is  assumed  as  01.  To 
predict  the  external  characteristic  for  a  lagging  current  the  con- 
struction is  as  indicated  by  the  full  lines  in  Fig.  25.  For  a 
leading  current  of  the  same  power  factor  the  construction  is  as 
shown  by  the  dotted-line  construction.  Because  of  the  simi- 
larity of  the  construction  to  that  used  for  a  noninductive  load, 
no  further  description  is  regarded  as  necessary. 


56  ALTERNATING  CURRENTS 

Expressed  analytically  in  this  case  the  regulation  is  given  by 


V(OEt  cos  0  +  IRY  +  (OEt  sin  0  +  IX)2  -  OEt   •*-  OEt 


It  will,  of  course,  be  found  that  the  lagging-load  current  gives 
a  much  higher  value  of  regulation  than  when  cos  0  =  1. 

If  the  regulation  is  desired  for  a  leading  current  (a  condition 
seldom  found  in  practice)  then  the  line  OEt  is  laid  off  below  the 
line  representing  current,  and  making  the  proper  angle  with  the 
current.  The  construction  is  otherwise  the  same;  solved  analy- 
tically the  regulation  is  given  by 


V(OEt  cos  0  +  IRY  +  (OEt  sin  0  -  IX)2  -  OEt  -s-  OEt, 


and  if  the  load  current  is  assumed  to  lead  more  than  a  few  degrees 
the  regulation  becomes  negative,  i.e.,  the  full-load  terminal 
voltage  is  greater  than  the  no-load  voltage  for  the  same  value 
of  field  current. 

The  entire  external  characteristic  may  be  calculated  from  the 
above  formulae  by  using  for  7,  instead  of  full-load  current,  that 
current  for  which  a  point  is  desired  on  the  external  characteristic. 

In  the  M.M.F.  method  the  magneto-motive  forces,  corres- 
ponding to  the  various  E.M.F.'s,  are  combined  vectorially.  The 
magneto-motive  force  for  a  given  coil  is  proportional  to  the 
current  through  it  and  so  the  current  through  a  coil  may  be  used 
as  a  measure  of  its  M.M.F.  If  the  magnetization  curve  of  the 
alternator  was  a  straight  line,  the  E.M.F.  would  be  directly 
proportional  to  the  M.M.F.  and  the  two  methods  of  prediction 
would  give  identical  results. 

In  the  M.M.F.  method,  the.  M.M.F.  necessary  to  overcome 
"  synchronous  impedance  drop  "  is  combined  vectorially  with 
the  M.M.F.  necessary  to  give  the  terminal  voltage  and  the  vec- 
tor sum  gives  the  total  M.M.F.  which  will  all  be  effective  in 
producing  E.M.F.  when  there  is  no  load  on  the  alternator,  i.e., 
no  "  synchronous-impedance  drop  "  to  overcome. 

The  M.M.F.  necessary  to  overcome  the  impedance  drop  should 
really  be  considered  in  two  parts,  that  to  balance  the  IR  drop 
and  that  to  balance  the  IX  drop.  The-grocedure  in  this  method 
will  be  understood  by  reference  to  Fig.  26.  The  phase  of  the 
current  is  given  by  the  line  OA  and  the  M.M.F.  necessary  to 
produce  full-load  reactance  drop  is  obtained  from  the  magnetiza- 
tion curve  and  is  then  plotted  as  OMX.  The  M.M.F.  necessary 


THE  ALTERNATING  CURRENT  GENERATOR      57 

to  produce  terminal  voltage  plus  IR  drop  is  plotted  as  OMt  and 
the  total  M.M.F.  is  their  resultant  OM'X.  By  reference  to  the 
magnetization  curve  the  total  induced  E.M.F.  (i.e.,  open-circuit 
voltage)  due  to  M.M.F.,  OM'X)  can  be  found  and  so  the  regula- 
tion calculated.  This  vector  problem  may  be  solved  analytically 


FIG.  26. 

in  the  same  manner  as  was  the  E.M.F.  diagram,  also  different 
power-factor  loads  may  be  dealt  with  in  the  same  way. 

Both  of  these  methods,  it  will  be  noticed,  treat  the  armature 
resistance  as  purely  ohmic,  and  treat  the  out-of -phase  component 
of  the  synchronous  impedance  as  due  entirely  to  reactance. 
As  a  matter  of  fact  the  effective  resistance  of  the  armature  is 
somewhat  higher  than  the  ohmic  resistance  due  to  the  "  load 
losses  "  (see  Experiment  15)  and  a  large  part  of  the  out-of -phase 
component  of  the  synchronous  impedance  is  not  reactance,  but 
represents  the  demagnetizing  action  upon  the  main  field  of  the 
armature  ampere  turns. 

A  method  is  here  given  which  considers  all  three  effects  which 
cause  a  change  in  the  terminal  voltage  with  change  of  load. 
These  effects  are  the  IR  drop  in  the  armature  (R  being  the 
effective  resistance),  the  IX  drop,  X  being  measured  with 
normal  excitation  in  the  field,  and  the  armature  reaction. 

With  such  field  excitation  as  gives  rated  terminal  voltage  on 
the  alternator  at  full  load,  and  the  armature  stationary,  impress 
upon  the  armature  an  E.M.F.  (from  some  outside  source)  of 
sufficient  magnitude  to  force  full-load  current  through  the  arma- 
ture. The  frequency  of  this  impressed  E.M.F.  is  to  be  that 
at  which  the  alternator  is  rated.  Read  current,  volts  and  watts 
input  to  the  armature.  The  reluctance  of  the  path  taken  by  the 
armature  reactance  flux  varies  with  the  angular  position  of  the 
armature,  as  described  in  Experiment  15,  hence  these  readings 
must  be  taken  for  several  different  angular  positions  of  the  arma- 
ture. Take  readings  at  six  positions  30°  (electrical)  apart, 
varying  the  impressed  E.M.F.  to  give  full-load  current  for  each 


58 


ALTERNATING  CURRENTS 


position  of  the  armature.  These  readings  are  averaged  and  the 
average  value  of  watts  and  voltage  used  to  calculate  Z  and  R  of 
the  armature.  Dividing  the  average  watts  by  rated  current  gives 
full-load  IR  drop  and  this  is  plotted  in  phase  with  the  current 
as  shown  in  Fig.  27.  With  a  radius  equal  to  the  average  im- 
pedance voltage  (from  above  test)  a  circular  arc  is  constructed. 

A  perpendicular  dropped  from 
R  to  this  arc  intersects  at  Z, 
and  OZ  is  then  the  full-load 
impedance  drop,  plotted  in  its 
proper  phase  relation  with  re- 
spect to  the  current.  Of  course 

>f    OX    gives    the    true    armature 

full-load  reactance  drop.  The 
normal  values  of  impedance,  re- 


O  R 

FIG.  27. 

sistance  and  reactance  obtained  from  this  test  will  be  called 
Z,  R  and  X. 

Now  with  about  J-normal  excitation  obtain  in  the  same  way 
other  values  for  these  three  quantities  and  call  them  Z',  R' 
and  X'.  These  values  will  be  used  in  connection  with  the 
synchronous-impedance  test  to  calculate  the  demagnetizing  effect 
of  the  armature. 

Now  using  the  value  of  full-load  synchronous  impedance  as 
determined  in  Experiment  12,  the  effect  of  armature  demag- 
netization is  obtained  by  vector 
diagram  as  shown  in  Fig.  28  The 
full-load  resistance  drop  is  plotted 
as  OR'  and  the  full-load  reactance 
drop  as  OX'.  These  values  of  Rr 
and  X'  are  for  the  low  excitation, 
OZ'  is  full-load  impedance  drop. 
Inscribe  an  arc  with  radius  equal 
to  full-load  synchronous-impedance 
drop  and  continue  R'Z'  to  inter- 
sect this  arc  at  V.  Construct  VY 
parallel  to  Z'O  and  VY  is  then  the 
maximum  value  of  the  armature  M.M.F.  measured  in  terms  of 
E.M.F.  at  low  saturation  of  the  field. 

Although  it  is  difficult  to  obtain  a  mathematically  exact 
formula  for  the  armature  demagnetizing  effect  the  following 
approximate  formula  may  be  logically  deduced,  D  =  Dm  sin  0, 


THE  ALTERNATING  CURRENT  GENERATOR      59 

where  Dm  represents  the  maximum  value  of  the  effect  and  <£  is 
the  phase  difference  between  the  armature  current  and  gene- 
rated E.M.F.  and  D  is  the  actual  demagnetizing  action.  We 

have,  therefore,  Dm  =  — — -  •     The  proof  of  this  relation  may  be 
sin  <p 

obtained  by  making  a  simple  assumption,  then  analyzing  the 
integral  effect  of  the  armature  currents  on  the  field.  The 
assumption  to  be  made  is  this  —  that  any  distributed  phase 
winding  can  be  represented  closely  by  a  concentrated  wind- 
ing having  the  same  number  of  ampere  turns.  For  instance, 
a  three-coil  per  phase  winding  of  one  turn  per  coil,  carrying  100 
amperes,  shall  be  represented  in  our  discussion  by  the  middle 
coil  only,  and  the  middle  coil  shall  be  supposed  to  carry  300 
amperes.  Such  a  redistribution  of  the  armature  coils  would, 
of  course,  mean  much  larger  reactance  per  winding,  but  we  are 
interested  in  this  discussion  only  in  the  M.M.F.  produced  by 
the  armature  ampere  turns  and  the  above  assumption  is  not 
far  from  the  actual  fact;  the  flux  produced  by  the  two  windings 
would  be  nearly  the  same. 

So  that  we  shall  suppose  on  the  armature  of  a  single-phase 
machine  only  one  coil;  a  three-phase  machine  will  have  three 


FIG.  29. 

coils,  etc.  Considering  a  bipolar  single-phase  machine  as  shown 
in  Fig.  29  and  designating  by  </>  the  angle  between  the  current 
and  generated  E.M.F.  in  the  coil,  we  have: 

i  =Im  cos  (a  —  $),    where   a  =  cot, 
therefore,  the  M.M.F.  of  the  coil, 

OM  =  KImcos(a-<j>). 

By  resolving  this  M.M.F.  in  directions  parallel  and  perpen- 
dicular to  the  axis  of  the  main  field  we  get  the  instantaneous 


60  ALTERNATING  CURRENTS 

values  of  the  demagnetizing  and  cross-magnetizing  effects  of  the 
armature  currents. 

Demagnetizing  action       =  OA  =  KIm  cos  (a  —  0)  sin  a; 

Cross-magnetizing  action  =  AM  =  KIm  cos  (a  —  0)  cos  a. 
To  get  the  complete  effect  of  these  actions  they  must  be  mi- 
grated throughout  a  cycle. 

Demagnetization  =  M.M.F.'  =  KIm  I     cos  (a  —  0)  sin  a  da. 

J-l 

/I 
COS  (a  —  0)cosada. 
it 

-2 

KI    r* 

Average  value  of  M.M.F/  =  —    -   I     cos  (a  —  0)  sin  a  da 

7T          *}  _7T 

KIm     C^  KIm     C^ 

=  —    -  I      cos  a  sin  a  cos  0  da  H  —    l   I     sin2  a  sin  0  da 

IT      J     IT  7T       J     IT 

~  ~ 


^  f* 

7T      «/_» 


cos  a  sin  a  cos*  AH-       = 

7T  _ 

n2  a  cos 


[sin'  a  cos  0]°_? 

a:  sin  </>lf      fsm  2  a  sm  01°         fsm  2  a  sm  ^1^ 

^~Jo~L  ~  ~J-rL   i~  J, 


In  the  single-phase  alternator,  therefore,  there  does  exist  an 
effective  demagnetizing  action  and  this  action  is  directly  pro- 
portional to  sin  0,  where  0  is  the  phase  displacement  of  generated 
E.M.F.  and  the  current  /  in  the  coil. 

It  is  also  interesting  to  notice  the  meaning  of  the  expression 

marked  (A);  while  the  coil  passes  from  position  —  ~  to  ~  this 

z        £ 

term  changes  its  sign  from  —  to  +.  In  the  single-phase  alter- 
nator there  is,  therefore,  the  pulsating  M.M.F.  which  tends  first 
to  weaken  and  then  to  strengthen  the  main  field.  This  action 
of  the  armature  produces  eddy-current  losses  in  the  pole  faces 


THE  ALTERNATING  CURRENT  GENERATOR      61 

of  the  machine  but  does  not  affect  the  average  strength  of  the 
alternator  field  to  a  noticeable  degree.* 

The  effective  value  of  the  cross-magnetization  in  the  single- 
phase  machine  may  be  obtained  in  a  manner  similar  to  that  used 
for  M.M.F/  We  have 

Tf 

KI    r^ 

M.M.F/' =  -     -   I      (cos2  a  cos  0  —  cos  a  sin  a  sin  0)  da 

7T     t/  _•* 

KIm  f  •   ,  i  a  cos  0   .  cos  2  a  cos  0~|  f 

= sin  a  sin  <f>  H 

TT    L  2  2  J_» 

=  J  E7m  cos  0. 

In  the  case  of  the  three-phase  machine  we  must  replace  the 
actual  distributed  phase  windings  by  three  "  equivalent  "  coils 
120°  apart.  Using  part  of  the  previous  expansion  we  may 
immediately  write  down  for  the  demagnetizing  action  of  a  three- 
phase  armature,  carrying  equal  currents  in  the  three  phases. 

J£T     C^1  C 
M.M.F/  =  — -  /      ]  [cos  a.  sin  a  cos  0  +  sin2  a  sin  0]  + 

7T       J  _7T     ( 

[cos  (a  4-  — }  sin  (a  -f-  —}  cos  0  +  sin2  (a  +  — }  sin  01  4- 
L      \        3 /       \        3  /  \3/         J 

\_cos(a      3/S3A         3/C<  1  \a      3/sm   1] 

T 
T£T  /»^  (  /  2  7T\  /  2  7T\ 

= I     cos0]  cosasina  +  cosfa +  -5-)sin(a -f -r)  + 

TT    J_|  V          «5  /        \  o  / 

/        ,     47T\     .       /        .     4w\)     , 

cos^+__jsm^+__j^a 

f     sin  0)  sin2  a  -f-  sin2(a+—  r)H~  sin2(a+-^)  (  da. 

_TT  (  \0/  \  6     /    ' 


Expanding  the  expression  in  the  bracket  of  the  first  integral  we 
get  cos  a  sin  a  =  cos  a  sin  a 

27T\     .      /        .     27T\  (  /I 


*  For  experimental  proof  of  the  presence  of  this  pulsating  armature  reac- 
tion, see  Appendix,  Plate  5.     Plates  16,  17,  and  19  also  show  the  same  effect. 


62  ALTERNATING  CURRENTS 

_  cos  a  sin  «  _  A/3  cos2  a      V3  sin2  a  _  3  sin  a  cos  a 

~^T~  ~T~  4  ~1T~ 

47T\      .       /  47T\          (/          1 


_  cos  a  sin  a   .   A/3  cos2  a  __  A/3  cos2  a  _  3  sin  a  cos  a 
4  4  4  4 

The  sum  of  the  right  hand  members  of  the  last  three  equations  is 
zero.  By  comparison  with  the  form  of  expression  obtained  for  the 
single-phase  alternator  we  see  that  the  first  integral  in  the  above 
expression  for  demagnetizing  action  is  that  one  which,  in  the 
single-phase  machine,  represented  the  pulsating  component  of 
the  demagnetizing  action.  As  this  integral  does  not  exist  in 
the  three-phase  machine  (owing  to  the  fact  that  the  above  sum 
reduces  to  zero)  we  conclude  that  there  is  no  pulsating  armature 
reaction  in  the  three-phase  machine. 

The  last  integral  must  still  be  evaluated  and  we  do  this  by 
first  expanding  the  various  terms. 

sin2  a  = 


sin 


>u         l 

cos  2 

OL 

27T\ 

3  ) 

2 

i2(«+' 

2 

">   \       1 

Z  TT  \           .I 

cos  2la  H 

3  /       2 

2 

M.M.F.'  =  Demag.  action  =  —  X  —  sin  0  =  I  /„  K  sin  <t>. 

7T  2  2 


THE  ALTERNATING  CURRENT  GENERATOR 


63 


By  an  exactly  similar  process  we  may  obtain  the  cross-magneti- 
zation and  shall  find  that  M.M.F."  =  f  KIm  cos  0. 

The  absence  of  the  pulsating  term  in  the  balanced  polyphase 
machine  might  have  been  surmised  when  it  is  remembered  that 
the  polyphase  currents  on  the  armature  produce  a  uniform 
magnetic  field  which  revolves  backward  on  the  armature  at 
synchronous  speed,  so  that  whatever  effect  it  produces  must  be 
stationary  in  space  because  the  armature  is  turning  forward  at 
synchronous  speed. 

Now  in  Fig.  29  it  is  evident  that  the  value  of  the  demagnetiz- 
ing action  in  the  synchronous-impedance  test  isZ'V,  and  that 


^ 

3 

/ 

/ 

A 

/H 

/ 

/ 

/ 

/ 

/ 

/ 

( 

1 

1 

/ 

, 

t 

1 

1 

O    D  E   F 

Magneto-motive  force 

FIG.  30. 


the  angle  between  generated  E.M.F.  and  current  is  0.  Hence 
the  maximum  value  of  the  armature  M.M.F.  (in  terms  of  volts  at 

Z'V 

low  saturation  of  field)  is  equal  to  VY  because  VY  =  — — -• 

sin  <p 

This  M.M.F.  is  now  to  be  changed  to  its  equivalent  in  volts  at 
normal  saturation  of  the  alternator  field.  Referring  to  Fig.  30 
(which  gives  the  magnetization  curve  of  the  alternator)  we  have 

0 A  =  D  (max.  demag.  action  in  volts  at  low  saturation) . 

OD  =  equivalent  M.M.F. 

OE  =  normal  excitation  of  field. 

EF  =  OD. 

Then  BC  =  max.  demag.  action  in  volts  at  normal  saturation 
of  field. 


64  ALTERNATING  CURRENTS 

The  vector  diagram  for  external  characteristic  prediction  is 
now  constructed  as  shown  in  Figs.  31,  32,  33,  which  are  for 
PF  =  1,  PF=  0.8  (leading)  PF  =  0.8  (lagging)  respectively. 

In  these  figures 

01  =  phase  of  load  current. 

OR  =  full-load  IR  drop  (effective  resistance  at  normal  satura- 
tion) . 

OX  =  full-load  IX  drop  at  normal  saturation. 

OB  =  maximum  demagnetizing  effect  in  volts  =  BC  of  Fig.  30. 

OEt  =  rated  full-load  voltage. 
OBB'  =  EgOI  =  <j>  =  phase  difference  between  generated  voltage 

and  armature  current  at  full-load. 
EgEg'  =  OB'  =  OB  sin  0  =  armature  demagnetizing    effect  at 

full  load. 

At  full  load  the  generated  voltage  is  OEg,  but  at  no  load  (the 
field  current  remaining  the  same)  the  generated  voltage  =  OEg', 


R  E.         E'      E? 

Non-inductive  load 

FIG.  31. 

because  EgEg',  the  amount  of  voltage  neutralized  at  full  load 
by  the  armature  reaction,  is  not  neutralized  at  no  load,  and  so 
appears  as  generated  voltage. 

As  the  load  decreases,  the  armature  demagnetizing  action, 
OB}  which  is  directly  proportional  to  the  armature  current, 
decreases  and  the  angle  0  also  decreases.  These  combined 
effects  give  for  the  locus  of  the  generated  voltage,  not  a  circular 
arc  through  Eg,  as  was  assumed  in  the  other  two  methods  of 
prediction,  but  the  curve  EgEt".  The  student  may  actually 
construct  this  locus  or  it  may  be  closely  approximated  by  first 
drawing  through  Eg,  an  arc  with  0  as  center.  This  arc  inter- 
sects the  line  assumed  as  the  phase  of  the  terminal  voltage  at 
E".  By  joining  Eg  to  Et"  with  a  curve  of  the  form  shown  in 
the  different  figures  the  locus  is  obtained  very  closely  without 
the  trouble  of  so  much  construction  as  is  necessary  to  get  the 


THE  ALTERNATING  CURRENT  GENERATOR 


65 


exact  locus.     Having  thus  obtained  the  locus  of  the  generated 
voltage,  the  method  of  predicting  the  voltage  for  any  load  is  the 


Load  with  Leading  Current 
FIG.  32. 

same  as  for  the  E.M.F.  or  M.M.F.  method.     To  get  the  terminal 
voltage  at  \  load,  e.g.,  take  OA  =  -=-,  draw  AC  parallel  to  OEt, 

draw  CEtf  parallel  to  AO  and  the  terminal  voltage  for  half -load 
is  given  by  OEtf. 

\ 


Inductive  Load 
FIG.  33. 

Using  the  same  alternator  as  was  tested  in  Experiment  11, 
and  12,  make  all  necessary  measurements  to  predict,  by  the  three 
methods  given  in  this  analysis,  the  external  characteristic.  Plot 
the  entire  characteristic  by  the  last  method,  for  loads  of  the 
same  power  factors  as  used  in  Experiment  11.  Indicate  on  the 
curves  so  constructed  the  points  actually  obtained  in  Experi- 
ment 11. 


66  ALTERNATING  CURRENTS 

On  another  sheet  plot  the  entire  characteristic  by  the  E.M.F. 
and  M.M.F.  method  for  the  same  power  factors.  By  compari- 
son with  the  results  of  Experiment  11,  see  whether  these  methods 
are  pessimistic  and  optimistic  as  stated? 

Calculate  from  results  obtained  by  these  two  methods  the 

,     .        .      /no-load  voltage  —  full-load  voltage\   .",".- 

regulation,  i.e.   -         ,  „  f  -  ,  for  loads  of 

\  full-load  voltage  /' 

all  three  power  factors  used  in  Experiment  11.  How  do  these 
results  compare  with  the  actual  test  results  and  with  those  ob- 
tained by  the  third  method  of  prediction? 


EXPERIMENT   XIV. 
EFFICIENCY  OF  AN  ALTERNATOR  BY  RATED   MOTOR. 

THE  most  direct  method  for  measuring  the  efficiency  of  an 
alternator  (or  any  generator)  would  be  to  actually  load  the 
machine,  obtain  the  output  by  reading  the  electrical  instruments 
in  the  load  circuit  and  measure  the  mechanical  input  by  means 
of  a  cradle  or  transmission  dynamometer.  Of  course,  this 
method  for  getting  the  efficiency  of  an  alternator  is  only  to  be 
used  with  small  machines;  with  large  machines  the  cost  of  the 
power  consumed  while  conducting  the  tests  becomes  important. 
Also  the  question  of  proper  loading  facilities  for  large  machines 
brings  up  difficulties  from  the  electrical  side  of  the  test  and 
the  measurement  of  the  power  input  by  dynamometers,  etc., 
presents  mechanical  difficulties.  So  for  testing  the  efficiency  of 
large  machines  one  of  the  various  "  loss  "  methods  is  used. 

For  small  machines,  however,  the  input-output  method  is 
simple  and  direct;  but  instead  of  using  a  dynamometer  to 
measure  the  input  we  may  drive  the  alternator  by  a  motor,  and, 
when  the  various  motor  and  transmission  losses  are  known,  the 
alternator  efficiency  can  be  calculated  by  reading  the  electrical 
output  of  the  alternator  and  electrical  input  to  the  driving  motor 
for  various  loads.  This  is  termed  the  "  rated  motor  "  method. 

When  a  certain  load  is  put  upon  the  alternator  the  input  to 
the  driving  motor  will  be  larger  than  the  alternator  output,  due  to 
three  general  losses  which  occur  in  the  combination  of  machines 
—  alternator  losses,  belt-transmission  losses  (this  factor  will 
be  zero  if  direct  coupling  of  machines  is  employed)  and  motor 
losses.  From  this  it  is  evident  that  if  the  motor  input  is  read 
and  motor  losses  and  transmission  losses  are  subtracted  from 
this  reading  then  the  difference  will  be  the  input  to  the  alter- 
nator. By  this  method  the  efficiency  formula  becomes: 

pffi  . alternator  output 

motor  input  —  (motor  losses  +  transmission  losses) 
+  alternator-field  loss. 
67 


68  ALTERNATING  CURRENTS 

The  losses  in  belt  transmission  and  their  variation  with  load  are 
not  easily  separated  from  the  alternator  losses  and  they  will, 
therefore,  be  assumed  constant  and  equal  to  (A)  *  per  cent  of  the 
alternator  input  at  no  load. 

The  motor  losses  consist  of  field  PR  loss,  independent  of  load, 
armature  PR  loss  varying  with  load,  and  stray-power  losses, 
involving  eddy-current  losses  and  hysteresis  in  the  armature  core 
and  pole  faces,  bearing  and  brush  friction,  windage,  etc.  These 
losses,  while  they  probably  do  change  somewhat  with  the  load, 
will  be  reckoned  constant.  Hence  the  only  variable  loss  to  con- 
sider in  the  motor  is  its  armature  PR  loss.  If  the  motor  used 
has  copper  brushes,  the  armature  resistance  may  be  treated  as 
constant,  but  when  carbon  brushes  are  used  the  resistance  of  the 
armature  circuit  changes  very  much  with  the  current  flowing 
through  it.  The  resistance  decreases  with  increase  of  current,  so 
that  if  the  armature  resistance  were  measured  at  a  small  value  of 
current  and  this  resistance  were  used  to  calculate  the  full-load 
PR  loss  in  the  armature,  much  too  high  a  loss  would  be  obtained. 
The  resistance  of  the  armature  must,  therefore,  be  measured  with 
various  values  of  current  from  small  values  to  125  per  cent  of 
full-load  current  of  the  motor.  From  this  data  a  resistance- 
current  curve  is  constructed  and  in  calculating  the  PR  loss  for 
any  particular  value  of  current,  the  proper  value  of  R  must  be 
obtained  from  the  curve. 

It  has  been  stated  that  the  hysteresis  and  eddy-current 
losses  in  the  motor  (for  a  constant  motor  speed)  may  be 
treated  as  a  constant.  This  statement,  as  well  as  the  one 
regarding  a  constant  PR  loss  in  the  field,  is  correct  because 
during  the  test  the  motor  field  current  is  to  be  maintained 
constant. 

But  the  alternator  must  be  run  at  its  rated  speed  for  all  loads 
and  the  motor  would  naturally  slow  down  as  its  load  increases. 
The  motor  speed  would  ordinarily  be  adjusted  by  weakening  its 
field  current,  but  the  field  current  is  to  be  kept  constant.  Hence 
the  armature  circuit  of  the  motor  must  contain  a  low,  variable, 
resistance  which  can  be  cut  out  as  the  motor  load  increases,  so 
that  the  alternator  speed  can  be  held  at  a  constant  value.  This 
resistance  should  give,  at  light-load  current,  a  voltage  drop  equal 

*  The  factor  A  varies  so  with  different  belting  that  no  definite  value  can 
be  assigned,  but  the  instructor  must  use  his  judgment  for  individual  instal- 
lations. 


THE  ALTERNATING  CURRENT  GENERATOR 


69 


to  about  10  per  cent  of  the  rated  voltage  of  the  motor  and  must 
be  capable  of  carrying  for  a  short  time  such  motor  current  as 
will  run  the  alternator  at  about  25  per  cent  overload. 

With  connections  as  designated  in  Fig.  34,  and  the  alternator 
running  light  (field  excited)  adjust  the  resistance  in  series  with 
motor  armature  until  it  uses  up  about  10  per  cent  of  the  impressed 
voltage  of  the  motor.  Then  adjust  the  motor  field  rheostat 
until  the  alternator  is  running  at  its  rated  speed  and  keep  the 
motor  field  current  at  this  value  throughout  the  test.  With 
the  alternator  running  at  rated  speed  get  motor  speed. 

Take  off  belt  (or  coupling),  run  motor  at  this  speed,  having  its 
field  current  at  the  value  just  determined,  read  input  to  motor 


•® — wvw 


D.C.  Supply 
O 


FIG.  34. 

armature.  These  readings  are  to  be  used  in  calculating  the 
alternator  input,  as  indicated  below.  After  these  readings  have 
been  obtained  the  alternator  is  again  connected  to  its  driving 
motor. 

Keeping  the  alternator  terminal  voltage  and  its  speed  con- 
stant, vary  its  load  (use  noninductive  load)  in  steps  so  as  to  get 
0>  f  >  i>  f  i  1  and  H  times  its  rated  load.  For  each  adjustment  of 
load,  read  volts  and  amperes  alternator  load,  amperes  in  alter- 
nator field  and  voltage  of  line  supplying  the  field  current,  motor 
field  current,  motor  armature  amperes  and  volts  and  alternator 
speed.  By  "  fall  of  potential  "  method  obtain  the  resistance  of 
the  motor  armature  for  several  values  of  current  covering  the 
range  of  current  used  by  the  motor  during  the  test. 

If        Wf  =  watts  input  to  motor,  alternator  disconnected. 

/i  =  amperes  input  to  armature  of  motor,  alternator 
disconnected. 


70  ALTERNATING  CURRENTS 

W"  =  watts  input  to  motor  at  some  particular  alternator 

load. 

1 2  =  amperes  in  motor  armature  alternator  at  the  same 
load. 

Raf  =  resistance    of    motor    armature,    alternator    dis- 
connected. 

Ra"  =  resistance  of  motor  armature,  alternator  loaded. 

Then  the  alternator  input  at  this  particular  load  will  be 
W"  -  I?Ra"  -  (W  -  I?Ra'}. 

Draw  armature  resistance  curve  using  current  as  abscissa. 
Calculate  alternator  efficiency  for  the  loads  used  above  and  con- 
struct efficiency  curve  through  the  points  obtained. 


EXPERIMENT   XV. 
EFFICIENCY  OF  AN  ALTERNATOR  BY  THE   "LOSS"   METHOD. 

THE  efficiency  of  any  machine  is  expressed  by  the  formula: 

Efficiency=iSr 

The  utilization  of  the  efficiency  formula  in  this  form  necessi- 
tates the  measurement  of  both  output  and  input  at  the  different 
loads  for  which  the  efficiency  is  desired.  As  this  method  is  many 
times  impracticable  and  expensive,  the  efficiency  of  a  machine 
is  generally  determined  by  use  of  the  formula: 

^rc  .  Output 

Efficiency  =  ^—.  — r^ 

Output  +  losses  in  machine 

It  is  seen  that  this  is  merely  a  different  way  of  putting  the 
same  formula,  because  input  =  output  +  losses. 

The  use  of  this  formula  for  the  determination  of  efficiency  is 
more  convenient  than  the  previous  one;  a  more  important 
reason  for  its  use  in  laboratory  tests,  however,  is  that  all  the 
different  losses  in  the  machine  must  be  separately  determined. 
Thus  the  action  of  the  machine  must  be  fully  analyzed  to  find 
out  how  the  different  losses  compare  in  magnitude,  how  they 
vary  with  the  load,  etc.  In  using  the  first  formula  given,  the 
action  of  the  machine  itself  need  receive  no  attention  whatever; 
the  only  measurements  involved  are  the  two  powers,  input  and 
output,  and  so  the  test  is  of  little  educational  value. 

In  the  "  loss  "  method  the  various  losses  are  plotted  in  the 
form  of  curves,  using  load  current  as  one  variable  and  losses  as 
the  other.  From  the  separate  loss  curves,  a  total  loss  curve  may 
be  obtained.  Then  for  a  certain  output,  the  efficiency  of  the 
machine  is  obtained  by  reading  from  this  curve  the  losses  for  the 
assumed  load  and  employing  the  formula: 

T^r-  .  Output 

Efficiency  — 


Output  +  losses 
71 


72  ALTERNATING  CURRENTS 

In  the  alternator  the  different  losses  are  field  PR  loss  and  field- 
rheostat  PR  loss;  armature  PR  loss;  windage,  bearing  and 
brush  friction,  hysteresis  and  eddy-current  losses,  all  grouped 
under  the  name  of  "stray  power;"  and  extra  iron  loss  designated 
the  "  load  loss."  Of  these  losses  the  stray  power  increases 
slightly  with  the  load,  the  field  and  rheostat  loss  increases  to  a 
slight  extent,  the  armature  PR  loss  increases  with  the  second 
power  of  the  load  and  the  load  loss  increases  with  some  power 
lower  than  the  second. 

As  the  alternator  is  supposed  to  be  a  constant- voltage  machine, 
and  as  it  has  been  shown  in  Experiment  12  that  the  field  current 
must  be  increased  to  maintain  constant  voltage  as  the  load 
increases,  it  is  quite  evident  that  the  field  loss  will  increase  with 
load.  If  the  armature  characteristic  of  the  machine  under  test 
has  already  been  obtained  the  curve  of  field  and  rheostat  loss  is 
readily  plotted  by  multiplying  the  values  of  field  current  by  the 
voltage  of  the  line  supplying  the  excitation  current.  Of  course 
the  loss  in  the  field  rheostat  must  be  charged  to  the  alternator 
as  it  is  one  of  the  power  losses  necessary  for  the  normal  operation 
of  the  machine. 

The  stray-power  losses  will  in  part  be  independent  of  load  and 
in  part  increase  with  the  load.  Air  friction,  bearing  and  brush 
friction  are  constant  quantities  within  the  limits  of  experimental 
accuracy;  the  iron  losses,  however,  will  slightly  increase  with 
load  because  the  speed  remains  constant  and  the  flux  density 
increases  with  the  load  due  to  increase  of  field  current.  In  this 
test  the  hysteresis  and  eddy-current  loss  will  be  considered  con- 
stant and  their  increase  will  be  measured  as  part  of  the  load 
losses. 

The  armature  PR  loss  is  ordinarily  obtained  by  multiplying 
the  square  of  the  load  current  by  the  armature  resistance  as 
measured  by  direct  current.  As  the  load  on  an  alternator 
increases,  so  does  the  leakage  flux  which  encircles  the  armature 
coils.  This  flux,  when  the  coil  side  is  in  the  interpolar  space, 
passes  through  a  short  path  in  the  armature  core  through  the 
teeth  of  the  armature  and  thence  completes  its  circuit  through 
air  as  illustrated  by  Fig.  35.  As  the  coi1  side  moves  under  a  pole 
the  pole  face  forms  part  of  the  path  of  this  flux  and  the  flux 
increases  because  the  pole  face  offers  less  reluctance  than  the  air 
path.  This  variation  in  the  magnetic  reluctance,  combined  with 
the  regular  alternation  of  the  current  causes  hysteresis  losses  in 


THE  ALTERNATING  CURRENT  GENERATOR      73 

the  teeth  (other  than  those  produced  by  the  main  field  flux 
variation  in  the  teeth).  Also  as  these  leakage  lines  utilize  the 
pole  face  for  part  of  their  path,  and  as  they  move  across  the  pole 
face  due  to  the  motion  of  the  armature,  hysteresis  and  eddy- 
current  losses  are  caused  in  the  pole  faces. 

As  a  matter  of  fact,  when  the  coil  side  comes  under  the  pole 
face  the  leakage  lines  do  not  exist  in  the  form  shown  in  Fig.  35, 
but  the  magneto-motive  force  of  the  coil  brings  about  a  shifting 


FIG.  35. 

of  the  main  field.  The  armature  ampere-turns  produce  a  cross- 
magneto-motive  force,  concentrating  the  main  flux  in  one  or  the 
other  pole  tip.  In  polyphase  alternators  this  cross  M.M.F.  is 
practically  constant  in  direction  and  magnitude  for  a  given  load 
current  so  that  in  this  type  of  machine  the  field  is  constantly 
distorted  in  one  direction;  in  the  single-phase  machine,  however, 
the  effect  is  an  alternating  one,  changing  in  magnitude  and  direc- 
tion. It  results  in  a  surging  of  the  field  flux  from  one  pole  tip 
to  the  other,  thus  producing  eddy  currents  and  hysteresis  losses 
in  the  pole  faces.  The  amount  of  surging  is  limited  by  the  eddy 
currents;  if  it  is  desired  to  keep  this  amount  of  surging  very 
small,  the  paths  for  the  eddy  currents  are  made  of  low  resistance 
so  that,  for  a  given  shifting  of  the  flux,  the  eddy  currents  are  as 
great  as  possible.  For  the  purpose  of  limiting  this  field  surging 
in  machines  where  it  may  produce  bad  effects  (as  in  synchronous 
motors  and  rotary  converters)  heavy  copper  grids  are  imbedded 
in  the  pole  face  to  furnish  low  resistance  eddy-current  paths. 

The  various  iron  losses  which  are  caused  by  the  armature 
current   are   grouped   under   the   name    "  load   losses."      The 


74  ALTERNATING  CURRENTS 

ordinary  methods  of  measuring  these  load  losses  are  purely  em- 
pirical, and  the  accuracy  of  the  results  obtained  is  much  to  be 
doubted. 

It  is  common  practice  in  laboratory  tests  to  short  circuit  the 
armature  and  excite  the  field  sufficiently  to  cause  full-load 
current  to  flow  through  it.  The  energy  used  up  in  the  armature 
and  pole  faces  is  measured  by  a  "  rated  motor."  The  ohmic 
PR  loss  in  the  armature  is  subtracted  from  the  value  obtained 
and  the  load  loss  is  reckoned  as  J  of  the  remainder.  The 
fraction  J  is  purely  arbitrary.  It  is  easy  to  see  that  the  full 
value  of  the  remainder  could  not  be  taken  as  the  load  loss  be- 
cause the  test  is  made  under  a  very  low  value  of  excitation,  under 
which  condition  the  armature  reaction  on  the  field  will  have 
much  larger  effect  than  when  the  field  is  more  nearly  saturated. 

It  is  proposed  in  this  test  to  modify  the  ordinary  method  of 
obtaining  armature  PR  loss  and  load  losses  by  a  method  in 
which  the  conditions  approach  actual  load  conditions  more 
closely  than  in  the  test  commonly  employed.  At  the  same  time 
the  increase  in  hysteresis  and  eddy-current  loss  due  to  increasing 
field  current  will  be  measured  and  included  as  part  of  the  arma- 
ture PR  loss  and  load  loss. 

The  method  here  proposed  for  obtaining  the  hysteresis  and 
eddy-current  losses  (of  constant  value)  and  the  armature  PR 
and  load  losses  will  probably  not  be  considered  commercially  im- 
portant, because,  although  very  little  power  is  required  for  the 
test,  an  A.C.  service  is  required  which  has  the  same  voltage  as 
the  machine  rating  and  a  current  capacity  equal  to  full-load  cur- 
rent of  the  machine;  generally  such  service  will  not  be  at  hand. 
The  method  is  applicable  to  laboratory  tests,  however,  and  is 
carried  out  as  follows: 

Adjust  the  value  of  alternator  field  to  normal  value,  no  load. 
Run  the  machine  as  a  synchronous  motor,  unloaded,  and  adjust 
the  impressed  voltage  until  the  P.F.  of  the  motor  is  unity.  The 
watts  input  under  these  conditions  gives  the  stray  power  + 
armature  PR  +  a  small  load  loss  which  will  be  included  with 
the  stray  power.  From  the  watts  input  under  these  conditions 
subtract  the  armature  PR  (ohmic)  and  call  the  remainder  stray 
power  which  will  be  regarded  as  constant  for  all  loads. 

Now  increase  the  field  current  of  the  machine  to  its  proper 
value  for  J  load,  as  obtained  from  the  armature  characteristic 
for  load  of  P.F.  =  1  in  Experiment  12.  Increase  the  impressed 


THE  ALTERNATING  CURRENT  GENERATOR      75 

voltage  until  J-full-load  current  is  reached  in  the  armature 
(machine  still  running  as  an  unloaded  synchronous  motor)  and 
read  the  watts  input.  Under  this  condition  the  armature  current 
will  lag  behind  the  impressed  E.M.F.  and  will  tend  to  magnetize 
the  alternator  field,  so  the  input  will  be  larger  than  it  should  be 
for  P.F.  =  1.  To  get  rid  of  this  error  in  the  measurement  leave 
the  value  of  alternator  field  at  its  proper  f-load  value,  and  de- 
crease the  impressed  E.M.F.  until  J-full-load  current  again  flows 
in  the  armature,  the  current  now  leading  the  impressed  E.M.F., 
and  read  the  watts  input  to  the  armature.  With  this  condition 
the  armature  current  will  tend  to  demagnetize  the  alternator 
field,  making  the  input  less  than  it  should  be.  Average  the  two 


FIG.  36. 

values  of  input  obtained  for  J  load,  subtract  the  "  stray  power  " 
and  the  remainder  will  be  (armature  PR  +  load  loss)  for  this 
value  of  load  current. 

It  has  been  shown  in  a  previous  experiment  that  the  armature 
reaction  of  a  synchronous  machine  will  magnetize  the  field  if  the 
current  leads  the  generated  E.M.F.  and  will  demagnetize  the  field 
if  the  current  lags  behind  the  generated  E.M.F. 

That  the  armature  current  does  lead  the  generated  E.M.F.  when 
the  alternator  (running  as  a  synchronous  motor)  is  generating 
an  E.M.F.  less  than  the  impressed  may  be  readily  shown  by  vector 
diagram.  That  the  current  taken  from  the  line  by  the  machine, 
when  the  line  E.M.F.  is  less  than  the  generated  E.M.F.,  lags  be- 
hind the  generated  E.M.F.  may  also  be  shown  by  vector  diagram. 


76  ALTERNATING  CURRENTS 

In  Fig.  36,  Em  represents  the  generated  E.M.F.  in  the  armature  of 
the  alternator  (running  as  a  synchronous  motor) .  The  impressed 
E.M.F.  is  given  by  E0  and  the  resultant  of  the  two  by  Er. 
The  load  on  the  motor  is  constant  and  is  supplied  by  an  elec- 
trical input  of  (Em  X  OA)  where  OA  represents  the  current  for 
that  value  of  impressed  E.M.F.  which  makes  the  input  current  a 
minimum.  If  the  impedance  and  armature  constants  are  known, 
Er  (the  voltage  necessary  to  overcome  the  armature  impedance 
drop  when  current  OA  is  flowing)  may  be  plotted  in  proper  phase 

and  magnitude.     If  Z  =  armature  impedance, 

•g 

and  tan  0  =  7^ 

Ka 

then  for  any  magnitude  and  phase  of  armature  current  the  neces- 
sary impedance  voltage  may  be  plotted. 

If  0  is  the  angle  between  the  current  and  Em  reversed  in  phase, 
it  is  evident  that  for  armature  current,  /0,  the  condition  which 
must  hold  is,  Emla  cos  0  =  constant. 

Hence,  if  in  Fig.  36  the  phase  of  Em  is  fixed,  the  locus  of  the 
armature  current  must  be  the  line  FD.  The  value  of  quarter- 
load  current  is  OB  (=  OC),  and  the  quarter-load  excitation 
produces  an  E.M.F.  =  Em'.  The  necessary  impedance  voltage 
is  Erf  and  the  impressed  voltage  is  either  OEaf  or  OEg".  Then 
with  a  motor  voltage  of  Em',  either  OEg'  or  OE0"  may  be  im- 
pressed on  the  motor  and  J-load  current  will  flow  through  the 
armature.  But  current  OC  will  demagnetize  the  field  and  the 
current  OB  will  magnetize  the  field  and  these  two  armature 
reactions  will  be  equal  in  magnitude  (as  0  =  0').  Hence  the 
input  with  impressed  voltage  =  OEgf  will  give  the  proper  PR 
loss  for  i  load  but  the  core  loss  will  be  less  than  it  should  be  by 
an  amount  =  X.  When  the  impressed  voltage  is  OEg"  the  PR 
loss  will  again  be  correct  but  the  core  loss  will  be  greater  than  it 
should  be  by  approximately  the  same  amount  X.  (This  reason- 
ing is  on  the  assumption  that  the  field  and  armature  are  not 
operated  near  the  saturation  point.) 

Hence  the  average  of  the  two  inputs,  when  the  impressed 
E.M.F.  is  OEgr  and  OEg"  will  give  the  proper  PR  loss  and  the 
proper  core  loss,  part  of  which  is  the  load  loss. 

Increase  the  alternator  field  to  its  proper  value  for  \  load. 
Increase  the  impressed  E.M.F.  until  J-full-Ioad  current  is  flowing 
in  the  armature  and  read  input.  Then  decrease  the  impressed 
E.M.F.  sufficiently  to  give  the  same  value  current  leading  the 


THE  ALTERNATING  CURRENT  GENERATOR      77 

impressed  E.M.F.  and  again  read  input.  Calculate  (armature 
PR  +  load  loss)  as  before.  Obtain  similar  readings  for  J, 
full,  and  1J  full  load. 

Measure  the  ohmic  resistance  of  the  armature.  Calculate  the 
armature  PR  loss,  and,  so  by  differences,  obtain  the  load  loss  for 
the  various  loads. 

Plot  on  one  sheet  of  section  paper  the  field  circuit  PR  loss,  the 
stray  power  loss,  the  armature  PR,  the  load  loss  and  also  the 
total  loss  curve,  all  curves  being  plotted  against  amperes  ouput 
as  abscissa.  Construct  the  efficiency  of  the  alternator  for  J,  J, 
J,  f,  full  and  1J  full  load  at  P.F.  =  1.  Compare  with  the  effi- 
ciency obtained  by  the  rated  motor  test.  Which  method  is 
the  more  accurate  and  which  is  preferable  for  testing  large 
machines? 


EXPERIMENT   XVI. 
PARALLEL  OPERATION  OF  ALTERNATORS. 

THE  question  of  series  or  parallel  operation  of  alternators  has 
become  important  with  the  present  practice  of  power  generation. 
Nearly  all  large  stations  generate  alternating  current  and  in  a 
single  station  are  installed  many  generators.  The  possibility 
of  operating  the  generators  as  separate  systems,  keeping  one  or 
more  feeders  for  a  certain  machine,  decreases  with  the  number 
of  generators  installed;  also  such  service  is  unreliable  and  difficult 
to  maintain.  For  reliability  of  operation  and  convenience  in 
making  repairs,  etc.,  it  is  best  to  have  all  machines  supplying 
power  to  the  same  bus  bars  and  all  of  the  feeders  connected  to 
the  same  station  bus. 

As  the  service  for  most  light  and  power  loads  is  required  to  be 
of  constant  potential  it  is  evident  that  the  alternators,  if  operated 
on  a  common  bus,  must  be  connected  in  parallel.  If  the  dif- 
ferent machines  were  connected  in  series  the  bus  voltage  would 
vary  with  the  number  of  alternators  supplying  the  load. 

Series  operation  of  alternators  on  ordinary  circuits  is  impossible 
unless  the  machines  are  rigidly  coupled  together,  because  if  the 
load  is  at  all  unbalanced  between  the  two  machines  they  immedi- 
ately pull  out  of  phase,  until  their  E.M.F.'s  are  acting  in  opposi- 
tion instead  of  in  series.  The  reason  for  this  fact  can  be  shown  by 
constructing  power  curves  as  shown  in  Fig.  37.  In  the  upper  figure 
are  given  the  E.M.F.  waves  A  and  B,  of  the  two  machines,  their 
combined  E.M.F.  acting  in  series  C,  and  the  load  current  D,  which 
is  taken  as  a  lagging  behind  the  E.M.F.,  C.  It  is  assumed  that 
the  two  machines  were  operating  with  their  E.M.F.'s  in  phase  and 
that  for  some  reason  B  began  to  lag  behind  A  a  little.  In  the 
lower  figure  are  shown  the  power  curves  for  the  two  machines, 
constructed  by  using  as  ordinates  the  product  of  the  current  D 
and  the  voltages  A  and  B.  It  will  be  seen  that  B}  the  machine 
which  is  assumed  lagging,  has  more  load  than  A;  this  unequal 
division  of  the  load  will  act  to  slow  down  B  still  more  and  allow 
A  to  speed  up,  making  the  distribution  of  load  still  more  unequal, 

78 


A.  C.   GENERATORS  IN  PARALLEL  OPERATION 


79 


until  B  is  supplying  whatever  load  there  is  and  also  supplying 
power  to  run  A  as  a  motor.  This  condition  occurs  when  the 
two  E.M.F.'s  are  nearly  180°  out  of  phase,  when  referred  to  each 
other.  As  they  pull  more  out  of  phase  the  line  voltage  C  de- 
creases until  finally  the  voltage  on  the  load  circuit  is  so  low  that 
the  load  supplied  by  the  two  machines  is  practically  nothing. 


Tower  Voltage 
Negative  Positive  Current 

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FIG.  37. 

It  is,  therefore,  evident  that  series  operation  of  alternators  for 
loads  where  the  current  lags  behind  the  generated  voltage  of  the 
machines  is  not  feasible,  the  operation  being  a  case  of  unstable 
equilibrium.  As  practically  all  commercial  circuits  furnish  a 
load  where  the  current  lags  to  some  extent,  it  is  seen  that,  even  if 
series  operation  of  alternators  was  desirable,  the  machines  could 
only  operate  successfully  if  mechanically  coupled  together.  If 
the  current  D  (Fig.  28)  was  in  phase  with  the  voltage  C,  the 
operation  would  not  be  unstable,  but  the  machines  would  not 
tend  to  maintain  any  special  phase  position  with  respect  to  each 
other;  any  phase  displacement  between  the  two  would  bring 
into  action  no  force  tending  to  hold  the  machines  in  the  same 
phase.  Hence  any  tendency  of  one  driver  to  get  ahead  of  the 
other  would  not  be  counteracted  and  the  relative  phases  of 


80  ALTERNATING  CURRENTS 

the  E.M.F.'s,  A  and  B,  would  change  at  random,  thus  causing  the 
line  voltage  C  to  vary  between  zero  and  a  maximum  equal  to 
the  sum  of  the  voltages  of  the  two  machines. 

By  means  of  a  vector  diagram,  the  question  of  stability  is 
easily  investigated.  In  Fig.  38  are  shown  the  three  possible 
conditions,  leading,  in-phase  and  lagging  load  current.  By  sup- 
posing machine  A  to  get  ahead  of  B  and  then  examining  the 
distribution  of  load  between  the  two  it  is  seen  that  (a)  with 


lagging-current  instability  exists;  (6)  with  in-phase  current 
indifferent  stability  exists;  and  (c)  with  leading  current  the 
operation  is  more  or  less  stable,  depending  upon  the  angle  of 
lead.  As  this  case  is  seldom  met  in  practice  it  has  little  com- 
mercial importance. 

Machines  operating  in  parallel  are  generally  quite  stable  and 
satisfactory  in  their  operation.  If  one  machine  speeds  up,  it 
immediately  takes  more  of  the  load  and  so  is  held  back.  A 
complete  analysis  of  the  factors  which  affect  the  parallel  opera- 
tion of  alternators  cannot  be  attempted  here,  but  only  those 
which  are  to  be  tested  in  the  laboratory. 

When  it  is  desired  to  connect  another  D.C.  generator  to  the 
line  to  which  power  is  already  being  supplied  by  other  machines, 
the  incoming  machine  is  merely  brought  up  to  line  voltage, 
tested  for  correct  polarity  and  switched  on  to  the  line.  It  will 
then  take  its  share  of  the  load  if  its  field  excitation  is  properly 
increased. 

To  switch  an  incoming  alternator  to  the  line  is  not  such  a 
simple  matter.  Before  the  switch  may  be  closed  three  condi- 
tions must  be  satisfied. 

1.  Machine  voltage  =  line  voltage. 

2.  Machine  frequency  =  line  frequency. 

3.  Phase  of  machine  E.M.F.,   180°  out  of  phase  with  line 
E.M.F. 


A.  C.  GENERATORS  IN  PARALLEL  OPERATION     81 

There  is  a  fourth  condition  which  must  be  fulfilled  before  the 
synchronizing  switch  may  be  closed,  but  as  this  condition  can- 
not be  affected  by  the  operator  it  is  not  grouped  with  the  other 
three,  all  of  which  are  adjustable  by  manipulation  of  rheostats, 
etc.  This  fourth  condition  requires  that  the  E.M.F.  wave  gene- 
rated by  the  alternator  be  similar  in  form  to  that  of  the  line 
E.M.F.  If  the  wave  forms  are  dissimilar  there  may  be  a  large 
interchange  of  current  between  the  machine  in  question  and  the 
other  machines  connected  to  the  line,  and  this  wattless  current 
circulating  in  the  local  path  does  no  useful  work  but  helps  to 
heat  the  alternators. 

The  cause  of  this  circulating  current  may  be  seen  by  reference 
to  Fig.  39,  in  which  the  line  E.M.F.  is  shown  by  the  full-line 
curve  A,  the  generator 
E.M.F.  by  the  dotted-line 
curve  B  and  the  difference 
between  the  two  by  the  full- 
line  curve  C.  The  line 
E.M.F.  is  supposed  to  be 
approximately  a  simple  sine 
wave  while  the  generator  is 
shown  with  a  fifth  harmonic. 
The  unbalanced  voltage  con-  FIG 

sists  principally  of  this  fifth 

harmonic,  shown  by  curve  C.  Now  it  is  to  be  noticed  that  the  two 
E.M.F.'s  are  such  that  both  give  the  same  reading  on  an  A.C. 
voltmeter,  i.e.,  they  have  the  same  effective  values,  and  it  is 
evident  that  this  unbalanced  voltage,  shown  by  C,  cannot  be 
reduced  to  zero  under  any  condition  of  field  adjustment,  etc. 
Now  this  voltage  C  will  force  a  current  to  flow  from  the  armature 
of  the  machine  in  question  through  the  armature  of  the  other 
machines  connected  to  the  line,  in  parallel.  This  circulating 
current  not  only  produces  unnecessary  heating  of  the  generators 
but,  if  of  appreciable  magnitude,  may  seriously  affect  the  stable 
operation  of  the  machine.  It  is  likely  to  produce  a  tendency 
for  the  generator  to  "  hunt,"  an  effect  which  is  explained  in  a 
later  experiment.* 

The  operation  of  closing  the  switch  which  connects  the  in- 
coming machine  to  the  line,  with  the  necessary  adjustments  of 

*  For  illustration  of  circulating  current  between  two  alternators,  see  Appen- 
dix, Plate  6. 


82 


ALTERNATING  CURRENTS 


Lamp 

-^V- 

, 

Line 

K 

(Iv 

Alternator 


voltage,  speed,  etc.,  to  satisfy  above  conditions  is  termed  "  syn- 
chronizing "  the  alternator. 

When  the  second  condition  is  approximately  fulfilled  (by 
speed  measurement  or  otherwise)  the  machine  voltage  is  made 
equal  to  the  line  voltage  by  field  adjustment.  Then  by  some 
sychronizing  device,  as  lamps  or  synchronscope,  condition  No.  2 
is  satisfied  more  closely.  By  watching  the  synchronizing  device, 
the  operator  can  tell  when  the  third  condition  is  fulfilled  and  the 
switch  is  closed. 

The  connection  of  lamps  to  be  used  for  synchronizing  is  given 
by  full  lines  in  Fig.  40.  The  lamps  used  must  each  be  built  for  a 

voltage  equal  to  that  of  the 
alternator  (e.g.,  a  220-volt 
alternator  requires  two  220- 
volt  lamps).  These  lamps 
complete  the  circuit  consist- 
ing of  the  alternator  arma- 
ture and  the  armatures  of 
the  other  alternators  already 
operating  on  the  line.  The 
voltage  acting  on  the  lamps  at  any  instant  is  equal  to  the  differ- 
ence between  the  line  voltage  and  machine  voltage;  if  they  are 
constantly  equal  and  opposite  this  voltage  is  zero  and  so  no 
current  flows  through  the  lamps  and  they  remain  dark.  But  if 
the  machine  frequency  is  not  equal  to  line  frequency  the  voltage 
on  the  lamps  varies  from  a  maximum  (when  the  alternator  and 
line  act  in  series  through  the  lamps)  to  zero,  and  so  the  lamps 
alternately  glow  and  become  dark. 

The  reason  for  the  intermittent  glowing  of  the  lamps  becomes 
apparent  when  a  vector  diagram  of  the  two  E.M.F.'s  acting  in  the 
local  circuit  is  shown.  Suppose  that  the  two  voltages  are  equal 
and  that  the  line  frequency  is  60  and  the  generator  frequency  is 
61;  then  the  vector  relations  of  the  two  E.M.F. 's  may  be  shown 
by  assuming  the  line  frequency  equal  to  zero  and  the  generator 
frequency  as  one  cycle  per  second.  In  Fig.  41  the  line  E.M.F.  is 
shown  at  OG.  The  generator  E.M.F.  is  shown  in  different  phase 
positions  with  respect  to  OG,  these  different  phase  positions  of 
generator  and  line  voltage  taking  place  as  the  generator  "  catches 
up  "  to  the  line  voltage  and  then  passes  it,  due  to  the  higher 
frequency  of  the  generator.  The  voltage  acting  on  the  syn- 
chronising lamps  is,  of  course,  shown  by  the  vector  OA,  OA' ', 


A.  C.  GENERATORS  IN  PARALLEL  OPERATION 


83 


OA",  OB,  etc.  The  locus  of  the  extremities  of  these  vectors 
will  evidently  be  a  circle  with  a  diameter  equal  to  the  sum  of 
the  two  E.M.F.'s.  So  that  the  E.M.F.  acting  on  the  lamps  passes 
through  all  values  be- 

E' 


\ 


FIG.  41. 


tween  the  sum  and 
difference  of  the  two 
E.M.F.'s,  a  number  of 
times  per  second  equal 
to  the  difference  of  the 
two  frequencies. 

In  the  middle  of  a 
dark  period  the  syn- 
chronizing switch  A 
may  be  closed.  Before 
the  switch  is  closed  the 
voltages  of  the  machine 

and  line  should  be  tested  for  equality  and  the  frequencies 
should  be  so  nearly  alike  that  the  period  of  the  lamp  is  approx- 
imately five  seconds.  If  the  lamps  are  connected  as  shown 
by  the  dotted  lines  in  Fig.  40,  the  proper  time  for  closing  the 
switch  A  is  at  the  middle  of  a  bright  period. 

In  general  the  alternators  will  not  be  of  so  low  a  voltage  that 
the  lamps  may  be  directly  connected  between  the  bus  and  ma- 
chine as  shown.  Where  the  voltage  is  higher  than  220  volts  a 
small  step-down  transformer  is  used  for  supplying  current  to 
the  lamp,  the  primary  of  the  transformer  being  connected 
where  the  lamp  is  shown  in  Fig.  40.  Instead  of  using  two 
transformers  and  two  lamps,  one  lamp  may  be  used  and  a 
small  transformer  fitted  with  two  high- voltage  windings  and  one 
low-voltage  winding  on  a  middle  leg  of  the  iron  core.  The  con- 
nections for  this  method  of  using  a  synchronizing  lamp  are  shown 
in  Fig.  42,  the  synchronizing  switch  being  shown  at  A  and  the 
transformer  and  synchronizing  lamp  at  B.  The  transformer 
is  so  connected  to  the  line  and  generator  that  when  the  two 
E.M.F.'s  are  in  opposition  practically  no  magnetic  flux  passes 
through  the  center  leg  of  the  core;  the  magneto-motive  forces  of 
the  coils  on  the  outside  legs  being  in  such  relative  directions  that 
nearly  all  of  the  flux  passes  around  the  core  through  the  outside 
legs  and  so  induces  no  E.M.F.  in  the  lamp  circuit.  If,  however, 
the  two  E.M.F.'s  are  in  phase  (i.e.  180°  from  the  proper  phase 
for  synchronizing)  then  the  direction  of  the  flux  is  as  shown  by  the 


84 


ALTERNATING  CURRENTS 


arrows  in  Fig.  42;  the  center  leg  serves  as  part  of  the  magnetic 
path  for  both  outside  coils.  The  lamp  will  be  bright  under  such 
conditions,  as  there  is  a  maximum  voltage  generated  in  the  coil 
of  the  center  leg  when  maximum  flux  passes  through  it. 

In  treating  this  question  of  parallel  operation  and  some  of 
the  factors  affecting  it,  the  case  considered  is  that  which  is  met 

in     commercial     generating 

^w  stations.  Several  alternators 

are  connected  in  parallel  and 
it  is  desired  to  determine  the 
behaviour  of  one  of  the  al- 
ternators when  either  its  ex- 


Line  2300 

, 
volts  \l 

Genere 

s. 

tor  2300  volts 

B 

•    ' 

f] 

'  ~ 

— 

1 

•  - 

-X- 

JL10  volt  lamp 
FIG.  42. 


citation  or  the  torque  of  its 
prime  mover  is  varied.  As 
the  machine  under  consider- 
ation is  only  one  out  of  a 
number  connected  to  the 

same  line,  it  is  to  be  first  noticed  that  neither  of  the  above  vari- 
ations can  affect  the  terminal  voltage  of  the  alternator,  this  being 
determined  by  the  rest  of  the  machines  in  the  station. 

With  this  condition  fixed,  a  vector  diagram  properly  con- 
structed serves  very  well  to  study  the  effect  of  variation  in  either 
the  excitation  or  the  torque  of  the  alternator.  In  Fig.  43,  OA 
represents  the  voltage  of  the  station,  to  be  designated  by  e.  A 
lagging  load  is  assumed  and  shown  by  01.  The  generated 
voltage  of  the  machine  under  consideration  is  equal  to  E  and  the 
armature-impedance  drop  equal  to  IZ.  By  using  0  and  A  as  cen- 
ters with  E  and  IZ  as  radii,  arcs  are  drawn,  intersecting  at  D. 
Then  DC  is  constructed  making  the  angle  a  with  DA,  where 

D 

cos  a  =  —~  and  Ra  =  armature  resistance,  wLa  =  armature  react- 
co-La 

ance  (including  demagnetization).  The  line  AB  is  drawn  par- 
allel to  01.  Then  this  diagram  represents  the  condition  when  the 
alternator  is  excited  to  voltage  E,  furnishing  current  I  and  gen- 
erating total  power  P. 

P  =  El  cos  <£'  =  El  sin  (6  +  a) 

=  El  (sin  d  cos  a  +  cos  d  sin  d), 


cos  6 


DF 
DA 


E  -  e  cos  8 


A.   C.   GENERATORS  IN   PARALLEL  OPERATION 


85 


AF 


e  smft 


(esin/3  cosa  -\-E  sma  —  e  cosjSsina). 
=  /o,  short-circuit  current  for 


excitation,  E. 

Therefore     P  =  I0e  (sin  ft  cos  a  —  cos  ft  sin  a)  -\-EIQ  sin  a 

=  IQ  e  sin  (ft  —  a)  +EIQ  sin  a. 


(1) 


FIG.  43. 


This  equation  serves  to  explain  the  fact  that  when  an  alternator 
is  first  connected  to  the  line  it  takes  no  load.  Before  syn- 
chronizing, the  operator  adjusts  the 
field  excitation  of  the  incoming  ma- 
chine so  that  its  voltage  just  equals 
the  line  voltage;  i.e.,  in  Fig.  43,  E  =  e. 
Also  the  synchronizing  switch  is  not 
closed  until  the  phases  of  the  line 
E.M.F.  and  machine  E.M.F.  are  oppo- 
site; that  is  in  Fig.  43,  ft  =  0°.  As  the 
torque  of  the  prime  mover  has  been  ad- 
justed, before  synchronizing,  to  just  sup- 
ply the  losses  in  the  incoming  machine, 
after  being  synchronized  there  is  no 
tendency  on  the  part  of  the  prime 
mover  to  change  ft.  But  when  ft  =  0°  and  E  =  e,  equation  (1) 
shows  that  the  electrical  power  generated  is  zero. 

The  angle  ft  is  the  phase  difference  of  the  alternator  E.M.F.  and 
line  E.M.F.  If,  after  the  incoming  machine  is  connected  to  the 
line,  the  torque  of  the  prime  mover  is  increased  it  tends  to  speed 
up  the  alternator  and  actually  does  so  for  a  fraction  of  a  second, 
i.e.,  the  alternator  pulls  ahead  of  the  line  E.M.F.  and  so  gives  to 
ft  some  value  other  than  zero.  As  the  power  that  the  alternator 
generates,  hence  the  necessary  torque  of  the  prime  mover, 
increases  with  increase  of  ft,  this  momentary  acceleration  of  the 
alternator  lasts  only  long  enough  to  give  ft  such  a  value  that 
(IQe  sin  (j3  —  a)  +  EIQ  sin  a)  equals  the  power  developed  by  the 
prime  mover. 

The  variation  of  load  on  an  alternator  as  ft  is  varied  is  readily 
determined  experimentally;  the  results  of  such  a  laboratory  test 


86 


ALTERNATING   CURRENTS 


are  plotted  in  Fig.  44,  which  gives  the  curves  of  phase  displace- 
ment with  variation  of  load  for  three  different  excitations  of 
the  alternator.  It  may  be  seen  from  these  experimentally  deter- 
mined curves  that  the  increase  in  power  output  of  the  alternator 
is,  within  experimental  error,  proportional  to  its  angular  position, 
i.e.,  the  value  of  (3  in  equation  (1).  It  will  also  be  noticed  that 
when  the  generator  is  overexcited  (E>e)  the  variation  of  /3,  for 


4000 
3000 

a 

i 

* 

5   2000 

1 

1000 
0 

^ 

/ 

<* 

/ 

- 

/ 

/ 

/ 

1 

A 

/ 

/ 

/ 

f 

/ 

/ 

j 

O 

/ 

/ 

/ 

/c 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

3 

f 

-, 

r 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

Value  of  field  current 
A  —  1.25  amperes 
B  —  1.02  amperes 
C  —    .8  amperes 

/ 

/ 

/ 

f 

J 

( 

y 

/ 

/ 

f 

/ 

/ 

/ 

/ 

/ 

/ 

f 

/ 

c 

/ 

i 

^ 

/ 

/ 

/ 

f 

/ 

1 

/ 

/ 

Variation  of  /? 
3  K.V.A.  Generator 

f 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

0°     4°     8°     12°     16°     20°     24°     28°    32°     36°    40° 

Phase  position  of  armature  in  electrical  degrees 
FIG.  44. 

a  given  increase  in  load,  is  less  than  on  normal  excitation  and 
for  subexcitation  (E<e),  this  variation  in  /5  is  greater.  This 
would,  of  course,  be  predicted  by  investigating  equation  (1). 
If  E  and  70  are  both  increased,  caused  by  increased  excitation  of 
the  alternator  being  tested,  a  correspondingly  less  increase  in  /8 
is  required  for  a  certain  value  of  P. 

If  the  prime  mover  does  not  exert  a  uniform  torque  (the  case 
with  all  reciprocating  engines)  the  value  of  /3  will  be  more  or  less 
variable.  The  frequency  of  its  variation  will  be  the  same  as 
the  frequency  of  variation  of  the  torque  of  the  prime  mover;  the 
amplitude  of  the  variations  of  J3  will  depend  not  only  upon  the 
variations  in  the  driving  torque,  but  upon  the  electrical  char- 
acteristics of  the  alternator.  If  /3  increases,  so  does  the  load  of 


A.   C.   GENERATORS  IN  PARALLEL  OPERATION  87 

the  alternator;  hence  a  great  increase  in  driving  torque  may  not 
produce  much  change  in  the  phase  position  of  the  alternator. 
In  the  curves  given  above,  it  is  seen  that,  for  normal  excitation, 
a  phase  displacement  of  20°  (electrical)  was  sufficient  to  change 
the  alternator  load  from  zero  to  full-rated  value. 

A  periodic  variation  of  0  (termed  hunting)  is  undesirable  be- 
cause under  certain  conditions  it  may  become  so  excessive  as  to 
throw  the  alternator  out  of  synchronism  with  the  line;  hunting 
is  also  likely  to  cause  disturbances  in  the  other  synchronous 
machines  on  the  line.  It  is  evident  that  to  keep  the  variation 
in  |8  small  for  a  given  variation  in  the  driving  torque,  the  alter- 
nator should  have  a  large  variation  in  power  output  for  a  small 
variation  in  )8.  From  equation  (1)  we  have: 


=  e/o  (cos  /3  cos  a  +  sin  0  sin  a) 

=  e!Q  cos  (/3  —  a)  . 

E  E  Ra  E    . 

Now      /o  =  —7=    ==  =  -  —  =  =  ~n~  sin  a. 


If  0  =  0 

dP      eE  Eesm2a 


This  expression  is  termed  the  synchronizing  power  of  an 
alternator.  It  is  a  measure  of  the  effort  of  the  alternator  to 
maintain  a  constant  phase  relation  with  respect  to  the  E.M.F.  of 
the  line.  It  will  evidently  be  a  maximum,  for  a  given  value  of 
Ra,  when  a  =  45°,  i.e.,  Ra  =  «La. 

In  a  good  commercial  machine  <oLa  is  much  larger  than  Ra. 
For  a  given  amount  of  copper  in  the  armature,  to  make  Ra  = 
coLa,  would  mean  low  efficiency  and  poor  regulation.  To  give  a 
large  synchronizing  power  to  a  machine,  therefore,  uLa  should 
be  decreased  as  much  as  possible  but  never  to  such  a  low  value 
that  a  <  45°.* 

*  This  idea  of  synchronizing  force  is  not  as  simple  as  indicated  above  when 
there  is  much  fly  wheel  effect  to  be  considered.  If  the  prime  mover  has  a 
variable  torque  and  the  rotating  member  has  a  large  fly  wheel  effect,  it  may 
be  advisable  to  make  the  synchronizing  force  very  low;  in  other  words  wLa 
should  be  very  much  greater  than  Ra- 


88 


ALTERNATING  CURRENTS 


We  next  investigate  the  effect  upon  the  machine  of  changing 
its  excitation  and  the  first  thing  to  consider  is  the  load.  Suppose 
that  E  —  e  and  that  (3  =  0,  which  condition  holds  just  after  the 
machine  is  synchronized.  The  machine  E.M.F.  and  line  E.M.F. 
may  be  shown  as  in  Fig.  45.  Now  suppose  that  E  is  decreased 
so  that  the  resultant  of  E  and  e  is  shown  by  OC.  The  current 
produced  by  OC  will  flow  through  the  armature  of  the  test 
machine  and  then  through  all  the  other  armatures  in  parallel. 
As  e  is  taken  as  line  voltage  it  is  evident  that  OC  must  all  be  used 


C'       0\  E          E'  E 


X 


FIG.  45. 

up  as  IZ  drop  in  the  armature  of  the  test  machine.  The  current 
/  will  lag  nearly  90°  behind  OC  because  of  the  comparatively 
high  inductance  of  a  good  armature  noted  above,  we  have 

IR 
tan  a  =  -=—£- ,  and  the  only  power  which  this  current  01  rep- 

/CO-La 

resents  is  El  sin  a.  But  this  is  the  power  lost  in  heating 
the  armature  due  to  the  resistance  loss.  It  is,  therefore,  evident 
that  increase  of  excitation  will  not  make  the  incoming  machine 
take  its  share  of  the  load. 

The  current  component  OX,  it  will  be  noticed,  is  wattless  and 
represents  no  power  output.  It  lags  90°  with  respect  to  OE  and 
hence  will  act  as  a  demagnetizing  current  for  the  overexcited 
alternator  and  will  tend  to  magnetize  the  other  machines  on  the 
line,  but  we  have  supposed  the  other  machines  to  be  of  such 
capacity  that  this  magnetizing  action  is  negligible. 


A.  C.  GENERATORS  IN  PARALLEL  OPERATION     89 

In  case  the  excitation  of  the  test  machine  is  decreased  the 
resultant  voltage  will  have  the  phase  OC1 '.  The  circulating 
current  is  O/',  having  a  watt  component  OR'  (supplied  to  the 
test  machine  by  the  line)  and  a  wattless  component  OX'.  The 
phase  of  OX'  is  opposite  to  that  of  OX]  it  will  tend  to  magnetize 
the  test  machine. 

A  complete  vector  analysis  of  this  question  of  variable  excita- 
tion is  possible  by  using  the  relations  expressed  in  equation  (1). 

P  =  IQ  e  sin  (|8  —  a)  +  EI0  sin  a. 

V 

Multiply  each  side  of  this  equation  by  -= — : —  and  add  to  each 

side  [— r —  )  • 
\2  sin  a/ 

Also  substitute  for  sin  (0  —  a)  its  equal  (—  cos  (90°  +  j8  —  «))• 
This  gives, 

'  (2) 


•  •  v>wo    i  t/v/          I       t~r  ***•  I  T          •  I    r\.        • 

sin  a/       sin  a  /o  sin  a     \2  sin  a 

Considering  the  right-hand  side  of  this  equation: 

pi 

We  have  just  seen  that  P  is  independent  of  excitation;  -=-  = 

armature  impedance,  which  is  supposed  constant,  and  we  have 
supposed  e  constant.     Also  sin  a  is  constant. 
We  may  therefore  write 

cos(90°  +  0-a)+(d^)2  =  ^  <® 


sin  a 
where 

7T2   _  if          ^          1     ._    •«•    \xl/o       I     Vw-^a/  /    _j 

"^       ~~     T     ~i_      "       I     I  O   ^^_  D 


=  ( 


70  sin  a        2  sin  a  fla  4  Ra* 

4 


+  wL«2)  J  (independent  of  E). 


Now  an  equation  in  the  form  of  (3)  represents  a  triangle  having 

P  P 

sides  E,  ^—   —.and  K,  and  the  angle  between  E  and  ^—  r  —  is 
7  2  sin  «'  2  sin  a 

(90°  +  |8  -a).  To  construct  the  triangle  erect  OA  (Fig.  46) 
perpendicular  to  the  base  line  and  equal  in  length  to  e.  Con- 
struct OB  at  an  angle  a  to  the  base  and  intersecting  at  B,  the 
line  constructed  perpendicular  to  OA  at  its  middle  point.  With 
a  radius  equal  to  K  and  center  at  B  construct  an  arc. 


90 


ALTERNATING  CURRENTS 


Through  0  draw  a  line  making  the  angle  0  with  OA  and  inter- 
secting the  arc  at  C.  Then  OBC  is  the  triangle  expressed  by 
the  equation  (3)  of  which  the  side  OC  is  the  generated  E.M.F. 
of  the  alternator.  As  OA  is  the  line  voltage  e,  the  line  AC  rep- 
resents the  impedance  drop  in  the  armature.  For  various  ex- 
citations the  point  C  will  lie  on  the  circular  arc  as  locus,  and  the 
current  will  be  proportional  to  the  length  oi  AC  and  constantly 
at  the  angle  (90  -  a)  to  AC. 


AI=miniT*mm  armature  current 
^<for  output;  BC;  Al/'for  BC/' 


etc. 


Loci  of  point  C, 
for  different  output 


FIG.  46. 

The  current  is  shown  at  A  I',  the  locus  of  I  is  easily  con- 
structed and  gives,  in  polar  coordinates,  the  variation  of  the 
armature  current  of  an  alternator  with  variations  of  generated 
voltage  OC. 

For  different  power  output  the  length  of  the  line  BC  has  a 
different  value,  but  the  rest  of  the  construction  is  identical. 
The  loci  of  the  current  are  given  in  Fig.  46  for  three  different 
loads.  They  show  that,  for  a  given  watt  output,  there  is  a 
certain  excitation  which  gives  a  minimum  current  output  and 
that  this  proper  excitation  increases  slightly  with  increase  of 
load. 

The  previous  discussion  holds  for  cases  in  which  the  test 
alternator  does  not  affect  the  line  voltage  or  the  magnetic  fields 
of  the  other  machines.  Two  alternators  of  approximately  the 
same  size,  operating  in  parallel,  will  not  give  results  exactly  as 
outlined  above  because  one  machine  influences  the  other  very 


A.  C.  GENERATORS  IN  PARALLEL  OPERATION     91 

much,  and  the  line  voltage  varies  with  the  excitation  of  either 
machine. 

It  is  a  very  interesting  and  instructive  test  to  actually  measure 
the  angle  0  (equation  1)  for  different  loads  and  excitations  of  the 
alternator,  in  order  to  see  whether  or  not  the  results  reached 
theoretically  may  be  experimentally  verified.  A  very  simple 
method  is  here  given  for  measuring  /3;  it  is  based  on  the  assump- 
tion that  whatever  manipulations  are  carried  out  on  the  alter- 
nator being  tested  the  field  distribution  of  one  of  the  alternators 
supplying  power  is  not  distorted  from  its  normal  position. 

It  will  be  remembered  that  our  investigation  of  the  effect  of 
armature  reaction  showed  a  cross-magnetizing  effect,  the  magni- 
tude of  which  depended  upon  the  current  being  carried  by  the 
armature  conductors  and  upon  the  power  factor  of  the  load. 
We  would,  therefore,  expect  a  shifting  of  the  field  of  the  alter- 
nators supplying  power  as  the  conditions  of  load  and  field 
excitation  of  the  test  machine  are  changed.  But  if  the  machine 
being  tested  forms  only  a  small  fraction  of  the  total  capacity  of 
machines  connected  to  the  line,  this  field  shifting  will  be  negli- 
gible. The  machine  used  for  obtaining  the  curves  given  in 
Fig.  44  had  a  capacity  of  3  k.v.a.  while  the  alternator  supply- 
ing it  with  power  was  of  30  k.v.a.  capacity  and  in  addition  had 
a  very  stiff  field. 

The  scheme  for  measuring  0  may  be  understood  by  reference 
to  Fig.  47.  Two  discs  of  some  insulating  material  are  used,  one 
on  the  shaft  of  the  machine  supplying  power  and  the  other  on 
the  shaft  of  the  machine  being  tested.  These  discs  are  fitted 
with  conducting  strips  running  from  the  center  conducting  drum 
to  the  outside  of  the  disc;  they  are  just  the  same  as  that  de- 
scribed in  Experiment  1,  used  in  the  plotting  of  curves  by  the 
method  of  instantaneous  contacts.  The  brush  bearing  on  the 
periphery  of  disc  A  (Fig.  47)  is  fixed  while  that  on  the  shaft  of 
the  test  alternator  is  movable  over  a  graduated  arc.  It  is  quite 
evident  that  the  lamp  will  burn  only  when  the  brushes  in  A  and 
B  both  make  contact  with  the  metallic  strips  at  the  same  instant. 
A  voltmeter  may  be  used  instead  of  the  lamp,  if  desired.  The 
two  discs  are,  of  course,  rotating  synchronously  (if  the  two  ma- 
chines have  the  same  number  of  poles),  so  it  is  evident  that  by 
shifting  the  brush  of  machine  B  a  position  may  be  found  such 
that  the  circuit,  containing  the  lamp,  is  closed  once  each  revolu- 
tion and  the  lamp  will  burn.  A  condenser  shunted  across  the 


92 


ALTERNATING  CURRENTS 


lamp  terminals  will  increase  the  brilliancy  with  which  the  lamp 
burns.  Why?  It  may  be  that  the  two  machines  do  not  have 
the  same  number  of  poles.  The  discs  work  most  effectively  when 
there  is  one  strip  for  each  pair  of  poles  of  the  machine  on  which 
it  is  mounted.  If  the  motor  is  a  six-pole  machine,  e.g.,  there 


110  volts 

OD.C.° 


Alternator  used  for 
phase  reference. 


x  voltage 
lamp  or  voltmeter 


Alternator 
being  tested 

FIG.  47. 


should  be  on  disc  B  three  strips,  spaced  120°  (mechanical)  from 
each  other. 

Suppose  the  movable  brush  of  machine  B  is  set,  with  no  load 
on  the  alternator,  so  that  the  lamp  burns.  If  now  the  torque  of 
B's  prime  mover  is  increased  it  will  be  found  that  the  lamp  no 


longer  burns;  the  movable  brush  of  B  must  be  shifted  forward  a 
certain  amount  before  the  lamp  is  again  bright.  The  amount 
that  the  brush  must  be  shifted  serves  to  measure  the  change  in 
angle  /3;  if  the  brush  position  for  /3  =  0°  is  desired  it  may  be 
found  by  finding  the  brush  position  to  make  the  lamp  burn  when 
there  is  no  exchange  of  power  between  machine  B  and  A  and 
when  B's  field  is  adjusted  to  give  an  E.M.F.  just  equal  to  the 
line  E.M.F. 


A.   C.   GENERATORS  IN  PARALLEL  OPERATION  93 

Set  up  apparatus  about  as  shown  in  Fig.  48;  using  lamps  or 
synchronoscope,  bring  the  alternator  into  synchronism  with  the 
line  and  note  the  current  in  both  field  and  armature  of  driving 
motor.  Close  switch  A  and  note  readings  of  all  instruments. 
Keeping  the  motor  field  at  the  value  just  read,  vary  the  alter- 
nator excitation  both  above  and  below  normal,  using  for  the 
limit  of  range  that  field  current  which  gives  25  per  cent  over- 
load current  in  A4.  Get  readings  of  all  instruments  at  about 
five  points  above  and  five  points  below  normal  alternator  field; 
read  also  phase  position  of  armature  for  each  value  of  field 
current. 

With  normal  field  (i.e.  the  field  current  which  makes  the  power 
factor  as  nearly  equal  to  one  as  possible;  this  current  will  be  dif- 
ferent for  the  different  loads)  on  alternator  increase  the  driving 
torque  (by  decreasing  A2)  in  steps  until  load  on  alternator  is  25 
per  cent  overload.  Get  about  six  readings  of  all  instruments, 
and  phase  position  of  armature,  in  the  range  given. 

With  half  load  and  with  full  load  on  the  alternator,  take  a 
series  of  readings  with  different  alternator  excitations  as  given 
above. 

Make  a  set  of  tests  using  instead  of  the  A.C.  line  another 
alternator  of  about  the  same  capacity  as  the  one  being  tested. 
With  normal  field  current  on  both  machines,  put  on 'the  line 
sufficient  load  to  bring  the  load  on  both  machines  up  to  rating. 
Adjust  the  prime  movers  to  make  the  alternators  divide  the  load 
in  the  ratio  of  their  capacities.  Adjust  the  field  excitations 
to  make  the  load  current  equal  to  the  sum  of  the  machine 
currents  (if  possible).  Then  leaving  all  adjustments  fixed, 
decrease  the  load  to  zero  in  about  five  steps,  reading  output  of 
each  alternator  and  load  current  and  line  voltage. 

It  will  be  noted  that  the  division  of  load  between  two  alter- 
nators under  these  conditions  depends  not  upon  the  shape 
of  the  external  characteristics  of  the  alternators,  but  upon  the 
form  of  the  speed-load  curves  of  their  prime  movers. 

Calculate  power  factor  of  the  alternator  for  all  readings. 

Plot  curves  as  follows  (for  tests  with  line  voltage  constant): 

On  one  curve  sheet,  with  alternator  field  current  as  abscissae, 
plot  alternator  armature  current,  watts  output,  and  power  factor 
for  the  three  runs  in  which  the  torque  of  the  prime  mover  was 
maintained  constant  and  the  test  alternator  was  connected  to  the 
constant  voltage  line. 


94  ALTERNATING  CURRENTS 

On  a  second  sheet  plot  the  relations  of  torque  of  prime  mover,* 
watts  output  and  alternator  field  current  to  phase  position  of 
alternator  armature,  using  phase  position  as  abscissae. 

On  a  third  curve  sheet  plot  the  results  obtained  from  the  test 
carried  out  with  alternators  of  equal  capacities.  Using  load  cur- 
rent as  abscissae  give  curves  of  watts  output  and  power  factor  of 
each  machine,  line  voltage  and  circulating  current.  Circulating 
current  may  be  obtained  by  use  of  the  formula 


Circulating  current  =  i//i  _  /IE)  where 

V  \E  1 


E 

W,  E  and  7  are  the  readings  of  the  wattmeter,  voltmeter  and 
ammeter  in  the  armature  circuit.     Explain  this  formula. 
Explain  all  results  using  vector  diagrams  where  possible. 

Note.  —  All  of  the  previous  tests  upon  the  alternator  have  supposed  a 
single-phase  machine.  If  desired,  some  of  these  tests  may  be  deferred  and 
run  on  a  polyphase  alternator,  after  the  preliminary  tests  on  polyphase  power 
have  been  performed. 

*  For  the  "  torque  of  prime  mover  "  may  be  used  the  product  obtained  by 
multiplying  together  the  armature  and  field  current  of  the  driving  motor. 
This  product  is  nearly  proportional  to  the  true  value  of  torque.  Why? 


EXPERIMENT   XVII. 

STUDY  OF  THE  CURRENT  AND  E.M.F.  RELATIONS  W  A  CON- 
STANT POTENTIAL  TRANSFORMER  ON  NO  LOAD  AND 
ON  FULL  LOAD. 

A  CONSTANT  potential  transformer  consists  essentially  of  two 
coils  of  wire,  insulated  from  one  another,  placed  on  a  laminated 
iron  core,  in  such  a  way  that  M,  their  coefficient  of  mutual  in- 
duction, is  as  nearly  as  possible  a  constant  maximum  value. 

If  a  sine  wave  of  E.M.F.  is  impressed  upon  one  coil  of  a 
transformer,  the  other  coil  being  on  open  circuit,  the  equation 
for  current  in  the  excited  coil  may  be  obtained  by  solving  the 
equation  which  expresses  the  relation  between  impressed  force 
and  reacting  forces: 

If         R  =  ohmic  resistance  of  coil  excited. 

F  =  the  field  set  up. 
E  sin  ut  =  impressed  force. 

x  =  current,  the  general  equation  will  be  of  the  form 

Esmut=Rx  +  jt(F). 

The  expression  Rx  gives  the  reacting  force  due  to  the  ohmic 
resistance  of  the  conductor  and  -^  (F)  expresses  all  of  the  re- 
actions caused  by  the  variations  in  the  magnetic  field.  In  this 
term  will  be  included  a  dissipative  reaction  due  to  hysteresis  and 
eddy  currents  in  the  iron,  a  dissipative  reaction  due  to  the 
secondary  current  (if  there  should  be  one)  and  a  nondissipative 
reaction  ordinarily  termed  the  transformer  reactance,  which  is 
caused  by  the  field  flux  cutting  the  conductors  of  the  primary 
coil. 

The  above  equation  is  not  in  a  form  which  is  solvable,  so  we 

must  express  -j  (F)  more  in  detail.     When  there  is  no  secondary 
current  the  equation  of  reactions  may  be  put 

Esmut  =  Rx  +  L^  +x^  (1) 

95 


96  ALTERNATING   CURRENTS 

where 

x  =  no-load  current. 

R  =  ohmic  resistance  of  primary  coil. 

L  =  coefficient  of  self-induction  of  the  coil. 

Now  L  is  defined  by  the  equation: 


where  S  =  number  of  turns  in  coil. 

A  —  area  of  magnetic  path. 
I  =  mean  length  of  magnetic  path. 
At  =  permeability  of  iron. 

TVT  -4  TrSxA/ji  /0, 

Now  0  =  --  j—  —  •  (3) 

By  investigation  of  this  formula  (3),  in  connection  with  a 
hysteresis  loop  as  given  in  Fig.  52,  it  is  seen  that  //  has  no  real 
meaning  when  the  iron  is  going  through  cyclic  changes  in  flux 
density.  When  the  magnetizing  force  is  zero  the  flux  still  has 
a  considerable  value,  giving  for  the  value  of  M,  °°-  Where  the 
hysteresis  loop  crosses  the  axis  of  M.M.F.  the  flux  has  zero 
value,  whereas  the  magnetizing  force  is  not  zero.  Here  the  value 
of  M  is  zero. 

A  sort  of  average  /z  may  be  obtained  experimentally  by  the 
use  of  formula  (3).  If  the  effective  values  of  4>  and  x  are  known 
from  test,  and  used  in  this  formula,  a  value  of  p  is  obtained 
which,  although  its  meaning  is  not  obvious,  may  be  used  for 
calculation  in  formulae  similar  to  (2).  But  such  a  value  of  ^ 
may  not  be  used  in  equations  involving  it  as  does  equation  (1). 

The  term  (  L  -rr  ~^~  x~jj]  must  be  such  a  function,  that,  integrated 

over  one  cycle,  it  gives  the  area  of  the  hysteresis  loop.  As 
neither  L  nor  x  are  expressible  as  simple  functions  of  the  time 
it  is  evident  that  the  solution  of  the  equation  is  not  possible 
without  making  some  assumptions. 

If  L  is  assumed  constant,  equation  (1)  may  be  solved  and  gives 

JF  T 

x  =     .        m  -  sin  (cot  —  8)  in  which  tan  0  =  -=-  • 
+  (coL)2  R 


In  commercial  transformers,  R  is  very  small  compared  with 
col/,  and  6  would  be  nearly  90°  if  it  were  not  for  the  core  losses 


THE  CONSTANT  POTENTIAL  TRANSFORMER  97 

which  have  already  been  mentioned.  The  hysteresis  and  eddy- 
current  losses  in  the  core  must  be  supplied  by  the  exciting 
current. 

These  losses  result  in  shifting  the  phase  of  the  current  with 
respect  to  the  impressed  E.M.F.,  so  that  in  practice  the  value  of 
6  may  be  between  70°  and  80°. 

In  equation  (1),  assuming  that  L  is  a  constant,  the  term  L-r- 
is  so  large  compared  to  R  that  it  may  be  put,  without  much 
error,  E  =  L  -=-  =  S  -^ ,  which  gives  the  angle  between  the  flux 

and  impressed  force  as  90°. 

The  investigation  of  the  reactions  in  the  transformers  from 
the  standpoint  of  differential  equations  will  not  be  carried 
further,  but  the  author  wishes  to  point  out  the  fact  that  the 
ordinary  treatment  of  the  transformer  using  the  differential 
equation  of  reactions  is  not  complete.  The  circuit  as  ordinarily 
depicted  does  not  provide  for  the  iron  loss  in  the  transformer, 
and,  moreover,  it  cannot  be  made  to  do  so.  If  a  complete 
mathematical  analysis  of  the  reactions  is  attempted,  the  trans- 
former circuit  must  be  imagined  as  made  up  of  two  primary 
circuits  instead  of  one.  This  is  shown  in  Fig.  49,  in  which  the 


FIG.  49. 

path  A  is  of  such  resistance  that  the  power  absorbed  in  it  is  just 
equal  to  the  iron  losses  in  the  transformer.  Then  the  line 
current  on  the  primary  side  is  made  up  of  two  components, 
/i,  in  phase  with  E  and  independent  of  load,  and  72,  which  itself 
will  consist  of  two  components;  the  one  component  will  serve  to 
magnetize  the  field  and  will  be  nearly  90°  behind  the  impressed 
E.M.F.  in  phase  (not  exactly  90°  because  of  the  ohmic  resistance 
of  the  primary  coil) ;  the  other  component  will  be  in  opposition 
with  the  secondary  current  and  will  be  of  sufficient  magnitude  to 
supply  a  M.M.F.  just  equal  and  opposite  to  that  produced  in  the 
secondary  coil  by  any  current  that  may  be  circulating  in  that 
coil. 


98 


ALTERNATING  CURRENTS 


Even  if  the  problem  is  attacked  (by  mathematical  analysis) 
with  the  above  represented  equivalent  circuit  in  place  of  the  real 
circuit,  an  exact  solution  cannot  be  reached  because  of  the  vari- 
ability of  L  (noted  above)  as  the  iron  goes  through  the  different 
parts  of  the  hysteresis  loop.  Also  in  the  actual  circuit  of  the 
transformer,  the  iron  losses  vary  to  a  slight  extent  as  the  load  is 
increased,  this  effect  is  not  accounted  for  in  the  circuit  of  Fig.  36. 

The  E.M.F.  in  the  secondary  coil  is  caused  by  the  variation  of 
the  flux  *  and  so  must  be  in  phase  with  the  primary  reacting  force 


x 


FIG.  50. 

caused  by  the  flux  variation.  But  at  no  load  this  reaction  is 
approximately  180°  out  of  phase  with  the  impressed  force,  so  that 
the  secondary  E.M.F.  is  also  180°  displaced  from  the  impressed 
E.M.F.  A  vector  diagram  showing  these  relations  is  given  in 
Fig.  50.  Here  the  effect  of  losses  in  the  iron  core  has  been 
neglected  and  also  the  IR  drop  in  the  primary  is  neglected. 


Actually  the  iron  losses  in  a  transformer  a?e  of  such  a  magnitude 
that  Fig.  50  does  not  represent  the  case  with  sufficient  accuracy 
and  the  relations  are  really  as  shown  in  Fig.  51.  The  exciting 
current  x  lags  behind  the  impressed  force  E  by  the  angle  0. 
The  component  x'  serves  to  magnetize  the  field  and  x"  supplies 
the  no-load  losses  of  the  transformer.  The  field  reaction  in  the 
primary  is  shown  by  +  ei,  at  right-angles  to  <£;  in  phase  with  e\ 
*  See  Appendix,  Plate  7. 


THE  CONSTANT  POTENTIAL  TRANSFORMER 


is  shown  e2,  the  secondary  E.M.F.  AB,  in  phase  with  x,  gives 
the  "  effective  resistance  "  drop  in  the  primary,  so  that  OB  must 
give  the  magnitude  and  phase  of  the  impressed  force  E.  It  will 
be  noticed,  by  comparison  of  Fig.  50  and  Fig.  51,  that  the  effect 
of  the  primary  resistance  is  to  slightly  change  the  ratio  of  in- 
duced secondary  E.M.F.  to  E.M.F.  impressed  upon  the  primary 
and  to  slightly  change  their  phase  relation.  The  first  of  these 
effects  is  very  small  when  the  transformer  is  not  loaded,  but  the 
second  is  easily  determined  experimentally. 

The  vector  analysis  of  the  currents  and  E.M.F. 's  when  the 
transformer  is  loaded  are  left  for  the  student  to  discuss. 

It  has  been  shown  that  if  the  magnetic  reluctance  of  the  trans- 
former is  constant  the  primary  exciting  current  (primary  current 
when  secondary  is  open  circuited)  is  of  the  same  form  as  the 
impressed  E.M.F.  In  the  actual  transformer  the  exciting  current 
is  far  from  being  a  sine  wave.  The  reaction  equation  of  the 
primary  circuit  when  the  resistance  component  is  neglected  (in 
discussing  the  current  form  this  will  be  done)  is 

.  d$f 

this  shows  that  the  flux  must  follow  a  cosine  wave,  0  =  0mcos  ut. 
Now  by  reference  to  the  curve  shown  in  Fig.  52,  giving  the 
cycle  of  magnetization  of  the  trans- 
former, it  is  evident  that,  if  the 
current  values  for  different  values 
of  0  on  the  cosine  curve  are  taken 
from  this  magnetization  curve,  the 
current  will  have  the  distorted  form 
shown  in  Fig.  53.  Not  only  is  the 
curve  distorted  but  it  reaches  its 
zero  value  ahead  of  the  flux  by  the 
angle  X.  The  value  of  this  angle  X 
depends  upon  the  width  of  the  hy- 
steresis loop  of  the  transformer  iron 
and  the  width  of  this  loop  is  a 
measure  of  the  energy  loss  in  the 
iron.  It  will  be  seen  that  this 
angle  X  brings  the  exciting  current  J^G.  52. 

more  into  phase  with  the  impressed 

E.M.F.,  as  it  should  do  to  represent  energy  consumption.  The 
result  of  these  differences  between  the  ideal  transformer  and  the 


100 


ALTERNATING  CURRENTS 


commercial  one  is  that  the  exciting  current  instead  of  being  a 
sine  current  lagging  90°  behind  the  impressed  E.M.F.  has  a 
peculiar  peaked  form  and  its  zero  value  lags  considerably  less 

than  90°  behind  the  zero 
of  the  E.M.F.  wave. 

When  the  secondary 
circuit  is  loaded  (sup- 
pose noninductively) 
then  the  current  flowing 
in  the  secondary  will  be 
in  phase  with,  and  of 
the  same  form  as,  the 
secondary  E.M.F.  By 
the  principle  of  conser- 


\\ 


FIG.  53. 


vation  of  energy  there 
must  be  an  equal  and 
opposite  current  fed  into  the  primary.  Then  the  primary  cur- 
rent will  be  the  resultant  of  this  load  current  and  the  exciting 
current.  As  the  exciting  current  is  only  a  small  fraction  of  the 
full-load  current  of  a  transformer,  when  there  is  any  appre- 
ciable load  on  the  transformer,  the  primary  current  loses  its 
distorted  shape  and  comes  very  nearly  into  phase  with  the  im- 
pressed E.M.F.  On  full  load  the  primary  current  shows  practi- 
cally no  distortion  and  is  displaced  only  a  few  degrees  from  the 
impressed  E.M.F. 

With  connections  as  shown  in  Fig.  54,  obtain  sufficient  points 
on  the  three  curves,  primary  and  secondary  E.M.F.  and  primary 
current,  to  get  their 
proper  forms.  Then 
put  full  load  on  the 
secondary  and  get  the 
four  curves,  E.M.F.  and  supply 
current  for  each  cir- 
cuit, both  for  inductive 
and  noninductive  loads. 

In  Fig.  54  the  leads, 
from  which  the  voltage 
curves  are  to  be  obtained,  are  shown  connected  across  one  of  a 
series  of  incandescent  lamps.  The  curve-tracing  apparatus, 
described  in  Experiment  1,  permits  the  tracing  of  alternating 
E.M.F.  waves  of  not  greater  than  110  volts  maximum;  the 


FIG.  54. 


THE  CONSTANT  POTENTIAL  TRANSFORMER  101 

transformer  shown  in  Fig.  39  is  supposed  to  give  a  110-110-volt 
transformation  so  that  the  curves  of  primary  and  secondary 
E.M.F.'s  could  not  be  measured  with  the  curve-tracing  appa- 
ratus; their  maximum  values  will  be  about  150  volts.  So  by 
connecting  two  lamps  in  series  and  taking  the  wave  form  of  the 
potential  difference  across  one  of  them  the  curve  obtained  is 
exactly  similar  to  the  E.M.F.  wave  desired,  but  of  one-half  the 
amplitude.  Some  such  scheme  as  this  is  often  necessary  when 
using  curve-tracing  apparatus  to  obtain  the  forms  of  high 
E.M.F.'s  or  currents. 

Caution.  —  Do  not  insert  much  resistance  in  series  with  the  primary 
circuit,  otherwise  the  exciting  current  will  not  have  its  characteristic  form. 
Why?  Could  you  measure  the  power  factor  of  the  exciting  current  from  the 
curve  sheet? 


EXPERIMENT   XVIII. 


REGULATION  AND  EFFICIENCY  OF  A  TRANSFORMER  BY  LOADING. 

THE  regulation  of  a  transformer  is  defined  as 

no-load  voltage  —  full-load  voltage 
full-load  voltage 

and  its  efficiency  by  - — — -  • 
input 

It  is  the  object  of  this  test  to  find  these  two  characteristics 
of  a  transformer,  and  also  the  power  factor,  by  loading  the  trans- 
former and  reading  the  proper  quantities. 

As  the  efficiency  of  a  transformer  is  very  high  (generally  over 
95  per  cent),  a  slight  inaccuracy  in  reading  or  in  calibration  of 

instruments  makes  a 
very  appreciable  error 
Load  in  efficiency.  If  there 
is  an  error  of  3  per 
cent  in  one  of  the 
meters  the  output  may 
be  read  larger  than  the 
input.  For  this  reason 
the  method  of  losses, 
to  be  given  in  the  next 
experiment,  is  much 
more  accurate  and  to 
be  preferred. 

Wire  up  the  trans- 
former to  be  tested  as 
indicated  in  Fig.  55. 


A.C. 

supply 


FIG.  55. 


When  the  same  meters  are  used  in  both  circuits  any  inaccuracy 
in  calibration  will  be  eliminated  in  calculating  efficiency.  By 
reading  the  wattmeter,  ammeter  and  voltmeter  in  the  secondary 
circuit  with  noninductive  load,  the  calibration  curve  of  the  watt- 
meter may  be  obtained  by  assuming  the  ammeter  and  voltmeter 
correct,  and,  of  course,  cos  0  =  1.  This  procedure  is  justifiable 

102 


THE  CONSTANT  POTENTIAL  TRANSFORMER  103 

because  in  getting  efficiency  the  same  wattmeter  is  used  in  both 
circuits  and  in  calculating  the  power  factor  absolute  accuracy  of 
the  wattmeter  is  not  necessary  but  it  must  be  accurate  with 
respect  to  the  product,  volt-amperes  as  obtained  from  the  am- 
meter and  voltmeter.  The  voltmeter  Vi  is  used  merely  for 
maintaining  the  impressed  voltage  constant  and  is  left  per- 
manently on  the  primary  circuit.  No  readings  should  be  taken 
until  the  impressed  voltage  is  exactly  at  that  value  which  gives 
rated  secondary  voltage  at  full  load,  noninductive. 

Adjust  the  impressed  voltage  so  as  to  give  rated  secondary 
voltage  when  full-load,  noninductive,  current  is  flowing,  and  keep 
this  value  of  impressed  voltage  throughout  the  test.  See  that 
the  frequency  is  kept  at  that  value  for  which  the  transformer  was 
designed.  Take  readings  of  input  and  output  for  values  of  out- 
put as  follows:  0,  J,  },  J,  J,  1  and  1J  rated  value. 

Take  similar  readings  for  an  inductive  secondary  load  of 
cos  0  =  0.8,  keeping  impressed  E.M.F.  the  same  as  it  was  for 
cos  <£  =  !. 

Calculate  efficiency  and  power  factor  of  primary  circuit,  for 
the  different  loads  measured.  Plot,  on  one  sheet,  efficiency, 
power  factor  and  secondary  volts  for  the  noninductive  load, 
using,  for  abscissae,  secondary  load  in  amperes.  On  a  second 
sheet  plot  similar  curves  for  the  inductive  load.  Calculate  the 
regulation  for  both  the  inductive  as  well  as  the  noninductive 
load. 


EXPERIMENT   XIX. 

EFFICIENCY,   REGULATION   AND   POWER  FACTOR   OF   A  TRANS- 
FORMER BY  THE  LOSS  METHOD. 

THE  two  losses  in  a  transformer  are  the  core  losses  and  the 
PR  losses  in  the  windings.  The  core  loss  is  practically  indepen- 
dent of  load  while  the  copper  loss  increases  with  the  second 
power  of  the  load. 

As  the  transformer  is  loaded,  leakage  of  the  flux  between  the 
two  coils  occurs.  This  leakage  flux  is  proportional  to  the  load 
and  it  so  produces  eddy  currents,  etc.,  that  it  causes  an  extra 
core  loss,  termed  the  "load  loss."  In  the  customary  method  of 
performing  this  test  the  load  losses  are  measured  with  the  copper 
loss  and  are  not  separated  therefrom. 

The  core  loss  consists  really  of  two  parts  which  may  be  sepa- 
rated from  one  another,  hysteresis  and  eddy-current  losses.  The 
hysteresis  loss  depends  only  upon  the  quality  of  the  iron,  the 
maximum  flux  density  and  frequency,  and  would  be  the  same 
if  the  core  was  made  laminated  or  solid,  i.e.,  this  loss  is  not 
affected  by  laminating  the  iron  except  that  thin  plates  of  iron 
may  be  better  annealed  than  solid  pieces.  To  keep  this  loss 
small,  special  quality  of  magnetic  steel  is  employed  and  the 
maximum  flux  density  used  is  comparatively  low. 

The  effect  of  the  wave  form  of  the  E.M.F.  impressed  upon  the 
core  loss  of  a  transformer  has  been  investigated  only  recently. 
It  is  found  that  if  two  E.M.F.'s  of  different  shapes  but  the  same 
effective  values  are  used  in  testing  the  core  loss  of  a  transformer, 
the  losses  will  be  different  in  the  two  cases.  It  is  stated  by  one 
writer  that  the  core  loss  of  a  transformer  may  vary  20  per  cent 
under  extreme  conditions  of  wave  distortion;  e.g.,  tested  with 
110  volts  (effective)  on  some  commercial  circuits  the  core  loss  of 
a  certain  transformer  might  be  recorder1  as  100  watts,  while,  with 
the  same  effective  voltage  from  a  different  circuit  of  wave  form 
differing  widely  from  the  first,  the  loss  might  be  as  much  as 
126  watts. 

A  peaked  E.M.F.  wave  will  produce  less  hysteresis  than  a  flat- 

104 


THE  CONSTANT  POTENTIAL  TRANSFORMER 


105 


50V 


E.M.F. 


topped  wave  of  the  same  effective  value.     This  is  really  due  to 

the  fact  that  the  hysteresis  loss  does  not  depend  upon  the  aver- 

age value  of  the  flux  density  reached  throughout  a  cycle,  but 

upon  the  maximum  value,  and  the  flat-topped  E.M.F.  wave  re- 

quires a  higher  maximum 

value  of  flux  than  does  a 

peaked  wave  of  the  same 

effective  value.     The  core 

loss  of  a  transformer  should 

be  guaranteed  only  in  con- 

nection with  the  wave  shape 

upon   which   it   is    tested. 

The  reason  for  the  greater 

maximum  value  of  the  flux 

when  the  E.M.F.  wave  is  flat-topped  may  be  seen  by  examina- 

tion of  the  reaction  equation  of  the  primary  of  the  transformer. 

Neglecting  the  resistance  reaction  (which  may  be  done  without 

much  error)  it  is  seen  to  be 


FlG. 


Now  we  will  consider  two  E.M.F.'s  waves  of  the  same  effective 
value,  namely,  100  volts,  as  shown  in  Figs.  56  and  57.     One  is 

a   rectangular   and   the 
other  a  triangular  wave. 


A  X 


Flux 


FIG.  57. 


wave 

has  a  maximum  value 
of  100  volts  and  the  tri- 
angular one  a  maximum 
value  of  172  volts. 

Using    the    equation 
given  above  we  have 


If 

K  J0 


edt. 


For  the  rectangular  wave  this  gives  <j>m 


100  TT 
2K 


For  the  triangular  wave  <f>m 


=i  n 

KJQ  \ 


2  X  172 


\tdt 


50  TT 
K  ' 

344  7T2 


43  TT 
K 


The  difference  in  <f>m  in  these  extreme  cases  is  seen  to  be  about 
14  per  cent,  and,  of  course,  the  hysteresis  loss  would  be  subject  to 


J 


106  ALTERNATING  CURRENTS 

a  much  greater  difference  than  this  as  it  varies  with  the  1.6th 
power  of  the  maximum  flux  density. 

The  eddy-current  loss  is  kept  down  by  using  laminated  iron 
for  the  core.  The  finer  the  laminations  the  less  will  be  the  eddy- 
current  loss,  as  will  become  apparent  by  considering  the  effect 
of  laminating.  In  Fig.  58  is  shown  a  cross  section  of  a  core  with 

laminations  much  thicker  than 
is  actually  the  case.  The  direc- 
tion of  the  flux  through  the  iron 
is  taken  as  being  perpendicular 
to  the  figure,  so  that  the  induced 
F!G  58  E.M.F.  in  the  iron,  due  to  flux 

changes,  will  be  in  the  plane  of 

the  figure.    The  path  of  an  eddy  current  is  shown  by  the  dotted 
line  in  lamination. 

The  magnitude  of  this  current  will  be 

E 


where  E  =  voltage  induced  in  eddy  path. 

R  =  resistance  of  eddy  path. 
X  =  reactance  of  eddy  path. 

The  loss  will  be  PR  =     ,  WR      -  (1) 


Now  only  those  paths  in  the  laminations  near  the  outside  of  the 
core  can  have  any  appreciable  reactance.  In  all  the  laminations 
inside  the  magnetic  field  of  the  core  the  current  in  the  eddy  path 
will  be  in  phase  with  the  time  variation  of  the  flux,  i.e.,  the 
reactance  of  the  paths  is  zero. 
We  may,  therefore,  put 

Loss  in  lamination  A  =  -^  • 
tt 

Now  in  laminations  B  (the  two  together  of  the  same  cross  section 

pi 

as  A)  we  shall  have  the  relations  E\  =  77- 

a 

RI  =  R  approximately  (apparent  from  diagram). 
Loss  in  both  B  laminations  =  2  Ii2Ri 


B,  R       2R 


THE  CONSTANT  POTENTIAL  TRANSFORMER  107 

So  by  finely  laminating  the  iron  the  eddy-current  loss  in  a  com- 
mercial transformer  is  kept  very  low,  being  only  a  small  fraction 
of  the  total  core  loss.  If  the  core  were  not  laminated  the  eddy- 
current  loss  would  be  many  times  the  hysteresis  loss. 

The  best  quality  of  iron  for  transformer  cores  is  one  having 
a  narrow  hysteresis  loop  and  a  high  ohmic  resistance.  It  must 
also  have  a  high  value  for  /*. 

The  core  losses  are  given  in  the  form  of  an  equation  as 

watts  (per  unit  volume)  =  KiBJ-'f  +  K2Bm2f2. 

Where      Bm  =  maximum  flux  density. 

/  =  frequency. 
KI  =  constant  depending  upon  area  of  hysteresis  loop 

of  iron. 

K2  =  constant   depending  upon   ohmic    resistance   of 
iron. 

The  copper  losses  (ohmic)  can  be  calculated  when  the  resist- 
ance of  each  coil  and  the  ratio  of  transformation  is  known. 
Each  (current)2  may  be  multiplied  by  its  respective  resistance; 
or  the  equivalent  resistance  Ri  +  azR2  (where  a  is  the  trans- 
formation ratio)  may  be  multiplied  by  /i2.  This  method,  how- 
ever, does  not  include  the  load  loss  and  so  the  copper  loss  is 
obtained  in  another  way. 

If  the  secondary  winding  is  short-circuited,  only  a  very  small 
E.M.F.  need  be  impressed  on  the  primary  to  force  full-load  cur- 
rent through  both  windings.  If  the  input  in  watts  is  measured 
under  these  conditions  it  will  be  the  full-load  PR  losses  +  a  cer- 
tain small  core  loss.  This  core  loss  is  probably  greater  than  the 
actual  load  loss  but  is  generally  used  as  the  load  loss. 

To  get  the  iron  loss,  have  a  suitable  ammeter,  wattmeter  and 
voltmeter  in  the  primary  circuit  and  the  secondary  circuit  open, 
then  impress  a  low  value  of  E.M.F.  upon  the  primary  and 
gradually  increase  it  until  rated  secondary  E.M.F.  is  reached. 
Adjust  the  frequency  and  voltage  of  supply  to  proper  rated 
value  of  the  transformer.  Read  current,  voltage  and  power 
supplied  to  primary  circuit  and  voltage  on  secondary  circuit. 
This  input  includes  a  very  small  I2R  loss  in  primary  coil  due  to 
the  exciting  current.  As  this  exciting  current  is  generally  less 
than  5  per  cent  of  rated  capacity  and  the  copper  loss  varies 
with  the  square  of  the  current,  the  copper  loss  for  the  exciting 


108 


ALTERNATING  CURRENTS 


current  is  of  negligible  value,  hence  this  input  is  taken  as  the  core 
loss  which  is  assumed  to  be  independent  of  load. 

The  reason  for  impressing  at  first  a  low  value  of  primary  E.M.F. 
and  gradually  increasing  it,  instead  of  throwing  the  primary  at 
once  on  a  line  of  normal  voltage,  is  the  protection  of  the  ammeter 
and  wattmeter  in  the  circuit.  With  the  secondary  circuit  open, 
after  the  steady  state  has  been  reached  in  the  primary  circuit,  only 
a  small  current  will  flow,  namely,  the  exciting  current.  As  this 
current  is  about  5  per  cent  of  the  full-load  current,  the  ammeter 
and  wattmeter  employed  in  this  test  will  have  a  current  capacity 
of  not  more  than  10  per  cent  of  the  transformer  rating. 

The  current,  which  flows  in  the  primary  circuit  for  a  short  time 
after  switching  to  a  circuit  of  normal  voltage,  however,  may  reach 

a  value  much  greater 
than  full-load  cur- 
rent.* This  excessive 
current,  although  only 
acting  for  a  few  cy- 
cles, is  quite  likely 
to  cause  serious  me- 
chancial  injury  to  the 
instruments.  The 
reason  for  the  rush  of 
current  is  given  by  the 
shape  of  the  magnetization  curve  of  iron  and  the  reaction  equa- 
tion of  the  primary  circuit.  Figure  59  gives  the  magnetization 
curve  of  the  core  iron  and  the  normal  hysteresis  cycle  for  the 
same.  The  maximum  flux  density  normally  reached  is  A. 

Now  it  may  be  that  the  power  is  switched  off  from  the  trans- 
former just  when  the  magnetism  is  at  value  A.  The  core  will 
start  to  demagnetize  along  the  upper  side  of  the  hysteresis  curve, 
and  the  flux  will  continue  to  decrease  until  some  point  B  is  reached. 
The  reaction  equation  of  the  primary  circuit  is  (when  the 
resistance  is  neglected), 

Impressed  E.M.F.  =  K^- 

Hence,  so  long  as  the  impressed  E.M.F.  is  positive,  from  A  to  B, 
Fig.  60,  -j-  must  be  positive,  i.e.,  0  must  be  increasing,  and  the 
total  change  in  <j>  necessary  to  balance  the  impressed  E.M.F.  from 
*  See  Appendix,  Plate  8. 


FIG.  59. 


THE  CONSTANT  POTENTIAL  TRANSFORMER 


109 


FIG.  60. 


A  to  B  (Fig.  60)  =  twice  the  maximum  flux  density  in  the  core 
during  normal  operation.  If  then  the  core  is  left  with  a  flux  = 
OB  (Fig.  59)  and  the  transformer  is  switched  to  the  supply  cir- 
cuit when  the  E.M.F.  is  zero  and  increasing  (time  A,  Fig.  60)  at 
time  B  the  flux  must  have  reached  a  value  OD  (Fig.  59)  where 
BD  =  20A. 

It  is  quite  evident  that  this  is  so  far  beyond  the  saturation 
point  in  the  iron  that  the  inductance  reaction  becomes  nearly 
zero.  Hence  the  impressed  E.M.F.  is  balanced  only  by  the  IR 
reaction,  which,  as  the  resist- 
ance of  a  transformer  is  very 
low,  necessitates  an  excessive 
value  of  current.  If  the  trans- 
former is  switched  to  the  sup- 
ply circuit  at  time  B  (Fig.  60) 
the  current  taken  during  the  suc- 
ceeding cycles  is  scarcely  more 
than  normal  value.  The  magni- 
tude of  the  switching  current 
may  be  anywhere  between  those 
given  by  the  two  conditions  cited,  depending  upon  the  flux  re- 
maining in  the  transformer  core  and  the  phase  of  the  E.M.F. 
when  switch  is  closed. 

With  the  secondary  short-circuited  through  an  ammeter  and 
suitable  ammeter,  voltmeter  and  wattmeter  in  the  primary, 
impress  a  very  low  voltage  on  the  primary.  Gradually  increase 
this  until  J-full-load  current  is  flowing  in  secondary  and  read  all 
instruments.  Increase  voltage  and  obtain  similar  readings  for 
J-,  f-rated  and  1  J-rated  secondary  current.  These  inputs  plotted 
with  secondary  current  as  abscissa  give  the  copper  loss  and 
load  loss  curve. 

These  two  losses  may  easily  be  separated;  the  ohmic  resist- 
ances of  the  primary  and  secondary  coils  may  be  obtained  by 
measurement  on  direct-current  test  and  these  resistances,  multi- 
plied by  the  square  of  the  currents  in  the  respective  coils,  give 
the  actual  copper  losses  of  this  transformer.  • 

As  a  transformer  is  generally  designed  the  copper  losses  in  the 
two  coils  are  equal,  so  that  if  the  copper  loss  is  calculated  for  one 
coil  the  total  copper  loss  may  be  taken  as  double  that  amount. 
However,  the  resistances  of  the  two  coils  must  be  measured,  as 
sometimes  the  above  relation  does  not  hold. 


110 


ALTERNATING  CURRENTS 


JQBSL 

i  A.C. 


After  the  ohmic  resistances  are  known  and  the  copper  losses 
calculated  and  plotted  in  the  form  of  a  curve,  the  load  losses  are 
shown  by  the  difference  between  this  curve  and  that  obtained 
in  the  A.C.  copper-loss  test.  This  is  shown  in  Fig.  61.  The 
load  loss  so  obtained  depends  upon  the  amount  of  magnetic 
leakage  between  the  two  coils;  if  the  coils  are  well  laminated  the 

load  loss  is  small,  but 
in  no  case  can  it  be 
zero;  unless  the  resist- 
ance of  the  coils  is  zero 
there  must  be  some  flux 
in  the  core  while  mak- 
ing the  A.C.  copper- 
loss  test;  this  flux  will 
give  a  load  loss.  It  is 
to  be  noted  that  this 
method  of  obtaining 
the  load  loss  is  purely 


90 

80 
70 

J  50 

I  40 

30 

20 

10 


3SS      / 

test/ 


10 


ralculaFed 


50 


20  30          40 

Load.Current 

FIG.  61. 

empirical  and  is  prob- 
ably far  from  being  accurate.  If  the  load  loss  was  an  important 
factor  in  obtaining  transformer  efficiency  some  better  method 
for  ascertaining  its  value  could  undoubtedly  be  formulated. 

The  different  losses  are  to  be  plotted  on  one  curve  sheet,  using 
load  current  as  abscissae. 

The  results  obtained  from  the  copper-loss  test  and  iron-loss 
test  may  be  used  for  predicting  the  behavior  of  the  transformer 
on  load. 

The  first  characteristic  to  be  predicted  is  the  curve  between 
secondary  terminal  voltage  and  load  current.  For  the  vector 
construction  are  needed  two  things:  the  open-circuit  secondary 
voltage  (with  normal  primary  voltage  impressed)  as  obtained 
from  the  iron-loss  test;  and  the  full-load  impedance  drop.  These 
factors  may  be  combined  as  in  the  simple  diagram  given  in 
Fig.  62.  OZ  represents  the  full-load  impedance  drop,  in  terms 
of  secondary  E.M.F.,  plotted  in  proper  phase  with  respect  to  the 
secondary  current-  01.  With  a  radius  of  E0,  the  open-circuit 
secondary  voltage,  an  arc  is  drawn  about  0  and  then  the  terminal 
voltage  at  full  load,  noninductive,  is  given  by  the  vector  OEt; 
at  half-load  by  OEt,  etc.  For  inductive  load  of  power  factor  = 
cos  6,  the  full  load  terminal  voltage  is  obtained  as  shown  at  OEt". 
The  terminal  voltage  for  any  load  whatever  is  obtained  by 


THE  CONSTANT  POTENTIAL  TRANSFORMER 


111 


subtracting  (vectorially)  from  OE0  the  proper  fractional  part  of 
the  vector  OZ. 

The  primary  power  factor  is  most  readily  predicted  by  the 
formula 

total  out-of-phase  current 


tan  0  = 


total  in-phase  current 


If  the  load  is  assumed  noninductive  the  only  out-of-phase  or 
wattless  current  is  the  magnetizing  current  while  the  watt  cur- 
rent is  equal  to  the  energy  component  of  the  exciting  current 


FIG.  62. 

plus  the  secondary  current  (changed  into  its  equivalent  primary 
current  by  the  known  ratio  of  the  transformer).  If  the  load  is 
assumed  inductive,  with 
power  factor  of  cos  0,  then 
the  use  of  above  formula 
for  tan  <f>  necessitates  the 
resolution  of  the  load  cur- 
rent into  its  two  compo- 
nents, and  each  must  be 
added  to  the  like  compo- 
nent of  the  primary  excit- 
ing current.  For  no  load 
the  power  factor  is,  of 
course,  that  obtained  in 
the  core-loss  test. 

The  next  characteristic 
to  be  predicted  is  the  efficiency,  and  the  method  employed  is 
made  clear  by  reference  to  Fig.  63,  which  shows  the  loss  curves 
and  the  curve  of  secondary  volts,  obtained  as  just  described. 


J.UU 

90 
SO 
70 

50 

30 
!  20 
10 

— 

—  — 

=; 

g 

ji? 

01 

d 

ir 

~ 

V 

"it 

s- 

^ 

= 

= 

-i 

^ 

^ 

/ 

^ 

^ 

x 

I1 

>t 

\\ 

0 

33 

^ 

^ 

--- 

•*• 

= 

—— 

^-* 

^ 

_ 

± 

JS 

L_ 

^4 

O         10          20          30         40          50          6 
Load  Current 

FIG.  63. 

112  ALTERNATING  CURRENTS 

At  any  assumed  load  OA,  the  loss  is  given  by  AB.  The  sec- 
ondary output  is  equal  to  OA  X  AC.  The  efficiency  is,  there- 
fore, obtained  as 

.  _          output  PAX  AC 

~  output  +  losses  ~  (OAXAC)  +  AB' 

Instead  of  using  vector  construction  to  obtain  the  external 
characteristic  and  power  factor  the  different  quantities  may  be 
combined  analytically  and,  if  very  accurate  results  are  desired, 
the  analytical  method  is  preferable. 


EXPERIMENT   XX. 

VARIATION  OF  CORE  LOSSES  AND  EXCITING  CURRENT  OF  A 

TRANSFORMER  WITH  VARYING   IMPRESSED   E.M.F.   AND 

FREQUENCY.     SEPARATION  OF  IRON  LOSSES  INTO 

HYSTERESIS  AND  EDDY-CURRENT  LOSS. 

ALTHOUGH  not  of  great  value  to  the  operating  engineer  the  data 
obtained  in  this  test  is  important  to  the  designer,  and  also  the 
test  serves  well  to  analyze  the  transformer  losses. 

The  watts  per  unit  volume  used  in  the  core  losses  are  given  by 
the  equation : 

Watts  =  KiBmi.«f  +  K2Bm*f* 

(Significance  of  symbols  is  given  in  Experiment  18.) 

The  value  of  the  hysteresis  exponent  1.6  holds  only  for  flux 
densities  up  to  10,000  gausses;  for  higher  densities  the  exponent 
becomes  greater;  for  very  low  densities  also  this  value  of  1.6  does 
not  hold.  The  value  of  K2  will  of  course  depend  upon  the  resist- 
ance of  the  eddy-current  path,  i.e.,  upon  the  core  temperature. 
It  is  probable  that  KI  also  depends  upon  the  core  temperature, 
but  there  seems  to  be  no  experimental  data  to  show  just  how  it 
varies. 

Examination  of  above  equation  shows  that  if  voltage  is  held 
constant,  the  core  loss  should  decrease  somewhat  with  increase 
of  frequency.  The  eddy-current  losses  will  remain  constant 
(for  a  given  impressed  E.M.F.,  B  varies  inversely  with  the  fre- 
quency) but  the  hysteresis  will  decrease  because  B  is  involved  to 
the  1.6  power,  while  /  is  only  to  the  first  power.  So  that  the 
total  core  loss  will  vary  inversely  with  /-5  (approximately) . 

For  a  given  frequency  the  flux  density  B  will  vary  directly 
with  the  impressed  voltage  (neglecting  the  resistance  reaction 
in  the  fundamental  equation  for  the  primary  circuit) .  The  eddy- 
current  loss  will,  therefore,  vary  directly  as  the  second  power  of 
the  voltage  if  the  core  remains  at  constant  temperature,  thus 
maintaining  Kz  constant.  The  hysteresis  loss  will  vary  with 
the  1.6  power  of  the  voltage.  The  total  loss  will,  therefore,  vary 
directly  with  the  impressed  E.M.F.  to  a  power  between  1.6  and  2, 
depending  upon  the  relative  magnitudes  of  the  two  losses. 

113 


114  ALTERNATING  CURRENTS 

The  exciting  current  will  be  directly  proportional  to  the  im- 
pressed E.M.F.  and  will  vary  inversely  with  the  frequency,  so 
long  as  the  reluctance  of  the  magnetic  circuit  remains  constant. 
If  the  circuit  becomes  more  or  less  saturated  then  the  current 
will  change  with  the  two  variables  to  a  power  higher  than  the 
first.  When  the  magnetic  circuit  becomes  saturated  a  further 
increase  in  the  E.M.F. ,  or  a  decrease  in  the  frequency,  will  make 
the  current  go  to  excessive  values  very  rapidly. 

For  separation  of  the  core  losses  into  the  two  components, 
only  two  measurements  of  the  watts  are  necessary,  keeping  the 
flux  density  constant.  When  the  core  is  operated  well  below 
the  saturation  point,  B  varies  directly  with  E.  Hence,  if  the 
core  losses  are  measured  at  two  different  frequencies,  keeping 
B  constant  by  keeping  the  ratio  (E.M.F.  +  f)  constant,  then  KI 
and  K2  of  the  loss  equation  may  be  found.  (If  two  different 
sources  of  E.M.F. 's  are  used,  care  must  be  taken  that  the  form 
factors  of  their  E.M.F.  waves  are  the  same.) 

Eddy-current  loss  is  the  same  for  all  frequencies,  when  im- 
pressed E.M.F.  is  constant,  but  varies  with  (E.M.F.)2  when  B  is 
maintained  constant,  so  we  may  put 

Eddy-current  loss  =  aE2,  where  (a)  is  a  constant. 

Hysteresis  loss  varies  with  the  frequency  and  with  the  (x) 
power  of  impressed  voltage  (x  will  ordinarily  be  1.6,  but  the 
following  demonstration  is  independent  of  its  value) . 

Hysteresis  =  bfBx  where  (6)  is  a  constant. 
Then  for  the  two  values  of  frequency  selected  we  have 
Wi  =  aES  +  &/!#«. 

Wz  =  aE22+  bf2Bx.     B  is  the  same  in  both  equations  when 
•p 

-r  is  held  constant. 

From  these  two  equations  the  eddy  current  loss  may  be 
computed. 

W1  -     -  Wz 
Eddy-current  loss  (at  Ei)  = 


::* 

This  loss  varies  directly  as  (voltage)2  and  is  independent  of  fre- 
quency, when  impressed  E.M.F.  is  constant,  hence  it  may  be 
readily  calculated  for  any  voltage  other  than  EI.  Hysteresis  loss 


THE  CONSTANT  POTENTIAL  TRANSFORMER  115 

is  then  found  by  subtracting  tke  eddy-current  loss  from  total  core 
loss. 

Find  the  core  loss  and  exciting  current  at  normal  voltage  with 
normal  frequency  and  values  },  li,  li,  normal  frequency.  (If 
such  variations  in  /  are  not  obtainable  use  four  other  values  in  a 
smaller  range.) 

Find  core  loss  and  exciting  current  with  normal  frequency  and 
values  of  voltage  f-,  f -rated  and  1  J-rated  value. 

Obtain  values  of  core  loss  at  normal  voltage  and  frequency  at 
one-half  normal  voltage  and  frequency,  from  which  calculate 
eddy-current  loss  at  normal  voltage.  Calculate  eddy-current 
loss  at  the  other  voltages  used  above  and  so  calculate  hysteresis 
losses  for  various  conditions. 

An  exponential  curve  such  as  Y  =  Xn  becomes  a  straight  line 
if  plotted  upon  section  paper  having  a  logarithmic  scale.  The 
tangent  of  the  line  gives  the  value  of  the  exponent. 

Plot  on  logarithmic  paper  curves  of  hysteresis  and  total  core 
loss  for  normal  voltage  and  varying  frequency.  Plot  similar 
curves  for  normal  frequency  and  varying  voltage.  By  measur- 
ing the  tangent  values  prove  that  hysteresis  varies  with  1.6 
power  of  voltage  and  first  power  of  frequency.  Upon  curve 
sheet  use  watts  for  ordinates  in  both  sets  of  curves. 

Plot  the  curves  of  losses  and  of  exciting  current  upon  linear 
coordinate  section  paper. 

If  a  transformer  designed  for  60  cycles  is  operated  at  133 
cycles,  what  could  you  predict  as  to  the  change  in  its  char- 
acteristics, the  supply  voltage  being  the  same  in  both  cases? 
What  if  operated  on  25-cycle  current  at  same  voltage? 


EXPERIMENT   XXI. 

HEAT   TEST    OF    A    TRANSFORMER    BY    OPPOSITION    METHOD; 
POLARITY  TEST. 

THE  rating  of  an  electrical  machine  is  determined  by  the  safe 
rise  in  temperature.  A  motor,  in  the  construction  of  which  is 
used  ordinary  insulation  (i.e.,  perhaps  cotton  and  shellac), 
might  be  rated  at  10  H.P.  The  rating  of  10  H.P.  means  that 
it  is  the  maximum  power  the  motor  can  deliver  without  over- 
heating some  of  its  parts.  If  now  asbestos  or  other  heat-proof 
insulation  were  used  instead  of  cotton,  the  same  size  motor  could 
be  rated  at  perhaps  15  H.P.  A  certain  percentage  of  the  input 
of  the  motor  is  used  up  as  heat  in  the  motor  itself;  when  this 
internal  loss  becomes  large  enough  so  that  the  heat  generated 
can  just  be  dissipated  by  the  machine  at  a  temperature  which  is 
safe  for  the  insulation,  etc.,  then  the  output  of  the  motor  under 
this  condition  should  be  its  rated  size. 

The  temperature-rise  test  is  not  taken  for  all  machines  sent 
out  of  a  factory  but  only  a  few  need  to  be  tested,  for  if  the  ma- 
chines have  the  same  electric  constants  (resistance,  etc.),  and 
have  the  same  radiating  surface,  they  will  behave  alike  in  their 
temperature  rise. 

To  determine  the  temperature  rise  of  a  machine  caused  by  the 
losses  which  occur  at  full  load  it  is  not  necessary  to  actually  load 
a  machine.  If  all  of  the  full-load  losses  are  being  generated  the 
temperature  rise  will  be  the  same  as  though  the  machine  were 
loaded.  So  different  methods  have  .been  devised  to  furnish  a 
machine  with  full-load  losses  without  actually  loading  it.  Gen- 
erally it  consists  of  testing  at  the  same  time  two  identical  ma- 
chines so  connected  together  that  one  has  generator  action  and 
the  other  motor  action,  the  result  being  that  each  machine  is 
operating  under  nearly  full-load  conditions  yet  the  line  is  fur- 
nishing merely  the  losses. 

In  the  case  of  two  transformers  this  is  accomplished  by  the 
"  opposition  "  method,  in  which  full-load  current  circulates 
through  each  transformer  thus  giving  full-load  copper  loss,  and 

116 


THE  CONSTANT  POTENTIAL  TRANSFORMER 


117 


'Variable  ratiTJ    i          ~~J   J~~ 
transformer 

O-i 


FIG.  64. 


rated  voltage  is  impressed  on  each  of  them  so  that  normal  core 
loss  is  generated.  The  method  consists  in  connecting  the  low- 
voltage  coils  in  parallel  and  impressing  normal  voltage;  the  sec- 
ondaries are  then  connected  in  series  (Fig.  64)  with  their  E.M.F.'s 
opposing.  Under  this  condition  normal  core  loss  occurs,  but 
in  the  primaries  there  is  only  the  exciting  current  and  in  the 
secondaries  there  is  no 
current  at  all.  If  now 
the  secondaries  are  opened 
and  a  source  of  variable 
E.M.F.  is  introduced  into  M 
the  circuit,  full-load  cur- 
rent may  be  caused  to  Q_ 
flow  through  both  trans-  A.C. 
formers  (in  both  coils)  by 
adjusting  this  E.M.F.  to 
a  value  equal  to  the  sum 
of  the  full-load  impedance 
drops  of  both  transform- 
ers. It  will  be  seen  that 

whereas  the  primary  coils  are  in  parallel  (i.e.,  opposition)  for  the 
outside  circuit  supplying  core-loss  current,  they  are  in  series  for 
the  E.M.F.  induced  in  them  by  the  current  which  is  caused  to  flow 
in  them  by  the  E.M.F.  introduced  into  the  secondary  circuit. 
Hence  full-load  losses  are  occurring  in  the  transformers  and  their 
temperature  will  rise  as  though  they  were  supplying  a  load  equal 
to  their  rating.  The  energy  supplied  is  just  equal  to  their  full-load 
losses,  hence  this  method  can  be  used  for  obtaining  efficiency  also. 
To  know  whether  or  not  the  secondaries  are  connected  in 
opposition  after  the  primaries  have  been  connected  in  parallel, 
it  is  necessary  to  make  a  polarity  test.  Although  the  terminals 
of  a  transformer  cannot  be  referred  to  as  positive  and  negative, 
as  they  change  polarity  with  the  reversal  of  the  alternating 
current,  yet  all  terminals  change  their  polarity  together  and  so, 
referred  to  one  another,  the  terminals  may  be  said  to  have 
polarity.  If  the  secondaries  are  so  connected  that  at  any 
instant  their  terminals  which  are  connected  together  have  the 
same  polarity,  then  no  current  will  flow  through  the  closed 
circuit  formed  by  their  coils.  If  they  are  connected  in  the 
opposite  manner  the  current  will  be  the  same  as  though  each 
were  short-circuited. 


118  ALTERNATING  CURRENTS 

To  test  whether  or  not  the  secondary  coils  are  in  opposition 
connect  together  two  of  the  terminals,  one  on  each  secondary, 
and  measure  the  voltage  between  the  remaining  pair  of  ter- 
minals. This  should  be  zero  if  in  opposition,  but  may  have  a 
small  value  due  to  a  slight  difference  in  the  ratios  of  the  two 
transformers.  The  voltmeter  used  must  have  a  range  equal  to 
twice  the  secondary  voltage.  With  high-voltage  secondaries 
where  it  is  not  convenient  to  use  a  voltmeter  the  circuit  may  be 
closed  through  a  long,  thin  fuse,  which,  if  the  coils  are  in  series 
instead  of  opposition,  will  blow  before  the  short- 

r\QQfiQfij '  circuit  current  can  do  any  harm  to  the  coils. 

I—  or, _   The  relative  polarity  of  a  primary  and  second- 

A  QOQOOOOQOOO  B 

ary  may  be  tested  by  connecting  as  an  auto 
transformer  as  in  Fig.  65.  If  the  voltage  across 
AB  is  less  than  the  voltage  BD  then  A  and  C  have  opposite 
polarity  and  should  be  so  marked. 

The  temperature  of  the  coil,  case  and  core  are  to  be  obtained 
by  thermometers  and  that  of  the  coils  by  resistance  measurement. 
On  the  core  and  case  fasten  the  bulb  of  the  thermometer  against 
the  iron  and  cover  with  a  piece  of  waste  or  putty  to  prevent 
radiation  from  the  bulb  and  so  that  it  may  register  the  true  tem- 
perature of  the  iron.  Hang  one  thermometer  in  the  oil,  having 
the  bulb  near  the  top,  as  this  will  be  the  hottest  part.  If  the 
thermometer  has  to  be  removed  to  be  read,  be  sure  to  put  the 
bulb  back  in  the  same  place  in  the  oil,  otherwise  very  irregular 
readings  will  be  obtained. 

If  the  transformer  has  been  standing  some  time  (say  24  hours 
for  a  small  one)  since  being  used,  the  coils  may  be  taken  to  be  the 
same  temperature  as  the  oil  when  beginning  the  test.  Measure 
their  resistance  before  current  has  been  passed  through  them 
long  enough  to  heat  them  appreciably.  Call  this  their  resistance 
at  the  oil  temperature.  At  any  other  resistance  their  tempera- 
ture t  may  be  calculated  by  the  formula  Rt  =  R0  (1  +  0.0038Z). 
Take  readings  every  ten  minutes  until  nearly  uniform  tempera- 
ture is  reached,  when  they  need  not  be  taken  so  frequently. 

If  the  rise  in  temperature  is  small  compared  with  the  absolute 
temperature  of  the  body  the  rate  of  radiation  may  be  assumed 
as  directly  proportional  to  the  difference  in  temperature  between 
the  body  and  the  surrounding  medium.  Under  such  conditions 
the  temperature  rise  will  be  a  logarithmic  curve  and  the  final 
temperature  rise  may  be  predicted  from  the  first  part  of  the 
curve : 


THE  CONSTANT  POTENTIAL  TRANSFORMER  119 

Assume     T  =  temperature  of  the  body. 
H  =  specific  heat  X  mass. 

A  =  radiation  per  degree  of  temperature  difference. 
K  =  rate  at  which  heat  is  supplied  to  body. 
TO  =  temperature  of  surrounding  medium. 
t  =  time  during  which  heat  is  supplied. 

H~  +  A(T-T0)=K. 


This  is  seen  to  integrate  into  the  form, 

J?r_™.*H 

in  which  K0  is  the  integration  constant. 


At 

~H. 

A       ~T~      •*•   0        I        -"-U^ 

When        t  =  0,          take  T  =  TQ,          then  KQ  =  ^ 

Therefore,  T  -  T0  =  -j(l  -  e" 

Therefore,  from  two  points  in  the  first  portion  of  the  temperature- 

xr 

rise  curve,  the  final  temperature  rise       :  may    be   predicted. 

Plot  temperatures  of  the  four  parts  measured  for  both  trans- 
formers using  time  as  the  abscissa. 

Caution.  —  After  a  resistance  reading  has  been  taken  be  sure  to  disconnect 
the  D.C.  voltmeter  before  the  D.C.  circuit  is  opened.  The  energy  stored  in 
the  magnetic  field  of  the  transformer  has  to  be  discharged  when  the  circuit 
is  opened  and  if  the  low-reading  voltmeter  is  across  the  coil  it  will  furnish  a 
path  for  this  energy  discharge  and  will  probably  be  injured. 


EXPERIMENT   XXII. 

STUDY  OF  THE  CONSTANT-CURRENT  TRANSFORMER  AND  DETER- 
MINATION OF  ITS  CHARACTERISTICS. 

FOR  ordinary  installations  of  power  the  supply  circuit  should 
be  constant  potential,  and  the  constant-potential  transformer  is 
designed  so  that  the  ratio  of  primary  to  secondary  E.M.F.  is 
practically  independent  of  load  current.  For  some  special  pur- 
poses (notably,  series  arc  lighting)  a  source  of  constant  alter- 
nating current  is  desired,  and  to  satisfy  this  requirement  the 
constant-current  transformer  has  been  designed.  On  a  series 
arc-light  system  when  the  number  of  lamps  varies  (e.g.,  some  of 
them  go  out  because  of  short  carbons  or  other  reasons)  the 
current  through  the  line  must  be  maintained  constant.  But  as 
the  number  of  lamps  in  the  circuit  varies,  the  E.M.F.  necessary 
to  maintain  constant  current  varies  also,  so  that  the  requirement 
of  the  arc-light  transformer  is  that  it  must  maintain  constant 
current  while  the  resistance  of  the  outside  circuit  is  varied  from 
zero  to  the  value  which  gives  rated  capacity  of  the  transformer. 
The  equation  of  reactions  in  the  two  coils  of  a  transformer  are  : 


at  at 


L  =  coefficient  of  self-induction  in  primary. 
R  =  effective  resistance  in  primary. 
x  =  current  in  primary. 
y  =  current  in  secondary. 
N  =  coefficient  of  self-induction  in  secondary. 
S  =  effective  resistance  of  secondary  +  outside  circuit 

in  series  with  secondary. 

M  =  coefficient  of  mutual  induction  of  the  two  coils. 
E  cos  co£  =  E.M.F.  impressed  on  primr  ~y- 

The  constant-current  transformer  is  made  with  the  coils 
movable  with  respect  to  one  another  and  it  was  found,  in  Experi- 
ment 5,  that  M  varied  with  the  relative  positions  of  the  coils. 

120 


THE  CONSTANT  CURRENT  TRANSFORMER  121 

When  M  and  L  are  constant  quantities  (as  in  constant-poten- 
tial transformer)  the  interpretation  of  these  differential  equations 
is  not  difficult  and  leads  to  the  ordinary  circle  diagram.  But 
when  M  and  L  vary  as  they  do  in  a  constant-current  transformer, 
the  relations  of  the  different  quantities  involved  can  best  be 
understood  by  analysis  of  the  vector  diagram  as  given  in  Fig.  66. 

The  flux  threading  both  coils,  when  secondary  is  open-circuited, 
is  represented  in  phase  by  $  and  the  primary  current,  to  produce 
this  flux  and  supply  hysteresis  loss,  by  OA.  Actually  this  flux, 
hence  the  current  OA,  will  change  somewhat  for  different  loads 
on  the  transformer,  but  as  OA  is  only  a  small  part  of  the  total 
primary  current  its  changes  will  be  neglected.  The  secondary 
impedance  is  assumed  constant,  which  is  another  questionable 
assumption.  However,  a  vector  diagram  constructed  upon  these 
two  assumptions  very  nearly  represents  the  actual  case. 

Suppose  when  M  has  its  greatest  value  the  transformer  has  a 
ratio  of  1:1.  The  voltage  impressed  on  the  primary  has  the 
constant  value  given  by  the  radius  of  the  circle  E£E\E\".  The 
current  to  be  delivered  by  the  secondary  has  the  constant  value 
given  by  the  radius  of  the  circle  Iz'Iz"-  Hence  the  primary  cur- 
rent, being  the  vector  sum  of  the  exciting  current  and  the  second- 
ary current  reversed  in  phase,  will  be  given  by  a  vector  from  0 
to  the  circle  /i/i",  which  circle  is  constructed  about  A  as  a  center 
and  radius  =  72. 

The  generated  secondary  voltage  (for  a  given  value  of  M)  is 
shown  by  the  vector  OE2.  If,  from  this,  the  secondary  imped- 
ance voltage  IzZz  is  subtracted,  the  vector  so  obtained  gives  the 
secondary-terminal  voltage  OE2.  As  we  have  assumed  a  non- 
inductive  secondary  load  the  IyZ2  drop  must  be  so  subtracted 
from  OE2  that  its  I2R2  component  is  in  phase  with  the  current  72, 
the  phase  of  which  has  not  yet  been  obtained.  The  method  of 
construction  to  satisfy  these  two  conditions  is  to  represent  I2Z2 
by  its  two  components,  then  to  change  the  position  of  I2Z^  until 
the  I2R2  component  if  projected  passes  through  0.  Having  now 
the  secondary  current  (magnitude  assumed  and  phase  just 
obtained)  the  primary  current  is  obtained  at  OI\.  The  three 
reacting  forces  in  the  primary  circuit  which  must  add  vectori- 
ally  to  give  the  impressed  force  are  the  I\R\  drop,  the  I\Xi  drop 
and  the  E.M.F.  given  by  OE2'  reversed  in  phase.  The  IiRi 
component  and  OE2  are  known  in  phase  and  magnitude  and 
1 1  Xi  is  known  in  phase  as  it  must  be  displaced  90°  from  OI\. 


122 


ALTERNATING   CURRENTS 


/ 


THE  CONSTANT  CURRENT  TRANSFORMER  123 

Combining  IiRi  and  OE2'  gives  OB.  To  OB  is  added  a  vector  at 
right  angles  to  the  current  OF.  This  vector  is  continued  until 
it  intersects  the  circle  Ei'Ei"  at  EI".  This  gives  the  phase  of 
the  primary  impressed  E.M.F.  and  so  the  problem  is  solved.  The 
power  factor  of  the  primary  circuit  is  given  by  cos  <f>. 

If  it  is  assumed  that  O/i  remains  constant  in  magnitude  (which 
it  will  do  approximately)  and  the  losses  in  the  transformer  are  con- 
stant (copper  losses  are  practically  constant,  iron  losses  increase 
slightly  with  increasing  load)  we  may  put  I2E2  =  I\Ei  cos  0  =  K, 
which  shows  that  as  E2  increases  the  power  input  to  the  trans- 
former must  correspondingly  increase;  as  I\  is  constant  this 
means  that  cos  <£  increases  directly  with  the  load,  which  will  be 
found  nearly  true  for  values  of  cos  <£  less  than  0.9. 

The  locus  of  the  secondary  terminal  volts  is  given  in  Fig.  66, 
and  it  is  seen  that,  throughout  the  working  range  of  the  trans- 
former, this  locus  is  nearly  a  straight  line. 

The  diagram  may  also  be  satisfactorily  constructed  by  suppos- 
ing that  all  of  the  transformer  inductance  is  in  the  primary  circuit; 
the  secondary  coil  having  resistance  only,  its  current  and  generated 
E.M.F.  should  be  then  considered  in  phase  with  each  other.  In 
many  ways  such  a  diagram  gives  a  more  logical  idea  of  the  trans- 
former quantities  than  the  one  given  here,  which  is  similar  to  the 
usual  transformer  diagram. 

Connect  the  primary  to  a  source  of  constant  potential,  of  volt- 
age and  frequency  the  same  as  transformer  rating,  and  put  in  the 
circuit  proper  instruments  for  measuring  the  power  input  and 
relative  phase  of  E  and  I.  In  the  secondary  circuit  put  an  am- 
meter and  voltmeter.  (If  an  incandescent  lamp  load  is  used  a 
wattmeter  is  not  needed.  If  arcs  are  used  for  load,  a  wattmeter 
will  be  necessary.)  A  potential  transformer  will  probably  be 
necessary  for  the  voltmeter  on  the  secondary  circuit. 

After  adjusting  the  counterweights  so  that  rated  current 
flows. in  secondary  at  full  load,  vary  the  load  from  zero  to  full 
load  in  about  eight  steps,  reading  all  instruments  and  height  of 
secondary  coil. 

Take  two  other  runs,  with  the  secondary  underbalanced  and 
overbalanced,  to  see  how  the  transformer  will  regulate  for  other 
than  rated  current.  Only  secondary-current,  position  and  sec- 
ondary E.M.F.  need  be  read  for  these  two  tests. 

Construct  a  vector  diagram  of  the  E.M.F.'s  in  the  two  circuits 
for  0,  J,  |,  J  and  full  load  on  secondary. 


124  ALTERNATING  CURRENTS 

Construct  curves  (with  watts  load  as  abscissae)  of  efficiency, 
power  factor,  secondary  E.M.F.  and  secondary  current  and  posi- 
tion of  secondary  coil.  On  another  sheet  plot  the  curves  of 
secondary  current  and  position  for  the  two  conditions  of  in- 
correct counterbalancing. 

How  will  the  different  losses  in  such  a  transformer  vary  with 
the  load  and  why? 

As  the  transformer  efficiency  is  not  very  high  and  the  power 
factor  is  very  low  at  low  values  of  secondary  voltage,  it  is  custom- 
ary to  use,  on  commercial  circuits,  transformers  having  several 
taps  on  the  secondary  coil,  so  that  the  transformer  may  be  oper- 
ated at  high  power  factor  on  different  circuits,  having  different 
numbers  of  lamps;  the  low  voltage  tap  is  used  for  a  circuit  having 
few  lamps,  etc. 


EXPERIMENT   XXIII. 

PARALLEL  OPERATION   OF  TWO   CONSTANT-POTENTIAL  TRANS- 
FORMERS. 

THE  conditions  which  affect  the  parallel  operation  of  alter- 
nators, i.e.,  wave  form,  equality  of  voltage  and  phase,  are  also 
present  to  affect  the  operation  of  two  transformers  in  parallel. 
In  the  case  of  the  alternators,  however,  the  attendant  can  regu- 
late the  conditions  to  reduce  the  cross  current,  etc.,  whereas,  if 
transformers  are  connected  in  parallel  to  one  feeder,  the  con- 
ditions cannot  be  readily  changed.  It  is  quite  evident  that  if 
the  transformers  are  to  operate  most  efficiently  there  should  be 
no  cross  current  exchanged  by  their  secondaries.  If  this  is  to  be 
so,  the  instantaneous  values  of  the  induced  secondary  E.M.F.'s 
minus  the  respective  impedance  drops,  due  to  load  current,  must 
always  be  equal,  otherwise  enough  cross  current  will  flow  to 
bring  about  such  a  condition. 

Considering  first  two  transformers  of  the  same  capacity,  it  is 
evident  that  for  such  condition  to  be  satisfied  their  respective 
impedances  must  be  equal  and  they  must  have  the  same  char- 
acteristic angle.  This  means  that  the  respective  resistances  and 
reactances  must  be  equal.  The  two  transformers  might  have  the 
same  value,  e.g.,  for  full-load  impedance  drop,  but  if  the  re- 
actance of  one  transformer  is  greater  than  that  of  the  other  (the 
corresponding  resistance  being  less)  then,  although  the  terminal 
voltage  of  the  two  transformers  will  be  of  the  same  magnitude 
when  full-load  current  is  flowing,  there  will  be  a  vector  difference 
between  the  terminal  E.M.F.'s  which  will  cause  a  cross  current 
to  flow.  Of  two  transformers  connected  in  parallel  the  one 
having  the  poorer  regulation  will  carry  the  smaller  share  of  the 
load. 

Now  if  it  is  desired  to  operate  in  parallel  two  transformers  of 
different  capacities,  the  full-load  impedance  drop  (not  impedance 
itself)  must  be  of  the  same  magnitude  and  phase  in  the  two. 

Connect  two  transformers  of  same  design  and  capacity  in 
parallel  and  vary  the  load  in  steps  of  about  one-fourth  of  the 

125 


126 


ALTERNATING  CURRENTS 


A.C.Supply 


combined  capacity  of  the  transformers.  Read  the  currents  and 
watts  output  furnished  by  each  and  load  (noninductive)  current. 

Obtain  the  impedance  and 
phase  angle  of  the  imped- 
ance drop  for  each  trans- 
former. 

Perform  the  same  tests 
for  two  transformers  of  dif- 

FlG  67  ferent  design  and  capacity. 

Calculate   the   value   of 

the  cross  currents  for  both  cases,  making  the  calculation  at 
every  one-fourth  increase  in  load  current.  If  connections  are 
made  as  in  Fig.  67,  it  is  seen  that  the  cross  current  may  be  ob- 
tained as  follows: 

Wi  =  watts  output  of  transformer  A . 
I1  =  total  current  output  of  transformer  A,  read  on  am- 
meter A. 
E  =  voltage  of  load  circuit,  read  on  voltmeter  V. 

Wi 

The  in-phase  current  furnished  by  transformer  A  =  -^-  •     The 

/          TWtf 
wattless  or  quadrature  current  of  A  =y  /i2—  \~w]  ' 

Two  3-K.W.  transformers  of  commercial  design,  supposed  to 
be  identical  in  all  respects,  gave  results  as  shown  in  the  curves 
of  Fig.  68.  A  and  B 
represent  the  currents 
supplied  by  each  trans- 
former to  the  load  cir- 
cuit and  C  is  the  cross 
current  circulating  be- 
tween the  two  second- 
aries. The  cross  cur- 
rent was  calculated  by 
the  formula  given  above 
and  so  represents  the 
wattless  current  only. 
But  the  true  cross  cur-  FIG.  68. 

rent   is  not   altogether 

wattless;  at  light  loads,  one  transformer  may  supply  consider- 
able energy  current  to  the  other.  This  occurs  in  the  above 


oss  current 
rrent  output,eachtransformei 

s  '  §  s  s 

~ 

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1 

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^ 

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^ 

t 

X 

^ 

-*" 

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X 

^ 

X 

-? 

t 

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£ 

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^ 

-" 

> 

X 

J 

^ 

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<s" 

s 

,^ 

-*• 

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X 

-^-- 

x11- 

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X 

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s-  3      v 

00-10 

10     .       20            30           40             50            00 

Load  current 

TRANSFORMERS  IN  PARALLEL  OPERATION  127 

curves,  when  the  load  was  18  amperes;  at  lower  values  of  load 
current  there  was  not  only  a  wattless  current  circulating  between 
the  two  secondaries  but  a  component  of  current  in  phase  with 
the  line  voltage.  The  dotted  portion  of  curve  C  represents  a 
cross  current  made  up  of  two  components,  a  wattless  current 
and  an  energy  current.  The  above  curves  show  the  be- 
havior of  two  transformers  which  had  slightly  different  ratios 
(giving  the  no-load  circulating  current)  and  also  different  im- 
pedance. By  adding  external  resistance  to  one  secondary  circuit 
and  some  inductance  to  the  other  these  two  transformers  were 
made  to  divide  the  load  equally  at  full  load,  but  the  equal 
division  was  not  maintained  when  the  load  varied.  The  no-load 
circulating  current  could  be  brought  to  zero  only  by  changing 
slightly  the  ratio  of  one  of  the  transformers. 

Explain  the  results  of  the  test  by  vector  diagram  where 
possible. 

If  time  permits,  add  either  inductance  or  resistance  to  the 
transformer  secondary  circuits  to  see  whether  or  not  the  load 
may  be  equally  distributed  and  whether  the  equal  distribution 
will  be  maintained  as  load  is  varied. 


EXPERIMENT  XXIV. 

TWO-PHASE  POWER,  UNIFORMITY  OF  POWER,  DIFFERENT 
METHODS  OF  CONNECTING,  VECTOR  ADDITION  OF  CUR- 
RENT AND  E.M.F.,  POWER  AND  POWER  FACTOR. 

A  POLYPHASE  system  is  one  in  which  are  combined  two  or  more 
single-phase  E.M.F.'s  differing  in  phase.  In  a  symmetrical  poly- 
phase system  the  different  E.M.F.'s  are  equal  and  equally  dis- 
placed from  one  another  in  phase;  in  a  balanced  polyphase 
system  the  loads  on  the  different  phases  are  equal  and  the  power 
delivered  by  such  a  system  is  uniform,  whereas  on  a  single-phase 
circuit  the  power  supply  is  pulsating. 

That  the  power  delivered  by  a  balanced  polyphase  system  is 
uniform  with  respect  to  time  may  be  easily  shown.  Consider 
a  two-phase  circuit,  the  voltage,  current  and  power  factor  of  each 
phase  being  the  same. 

Total  power 

=  Em  COS  0)t  Im  COS  (O)t  —  0)   +  Em  COS  I  at  +  ^  J  Im  COS  ( tot  +  |  — 

=  Emlm  (cos  cot  cos  (co£  —  0)  +  sin  ut  sin  (ut  —  0). 
This  gives  by  expansion : 

Total  power  =  Emlm  cos  4>, 

which  is  evidently  independent  of  time.  In  the  same  way  a  three- 
phase  circuit  may  be  shown  to  supply  energy  at  a  uniform  rate. 
The  advantages  of  the  polyphase  systems  over  the  single-phase 
systems  are  economy  of  material  in  machine  construction  (poly- 
phase machinery,  e.g.,  generators,  motors  and  transformers, 
have  less  weight  per  K.W.  capacity  than  single-phase  machines) 
and  economy  in  transmission-line  construction.  Also  certain 
machinery,  especially  induction  motors  and  rotary  converters, 
practically  require  polyphase  power  for  satisfactory  operation. 
The  single-phase  induction  motor,  has,  per  se,  no  starting  torque; 
the  single-phase  rotary  has  only  a  small  percentage  of  the 
capacity  of  the  same  machine  run  polyphase  and  its  operation 
is  less  satisfactory,  etc. 

128 


POLYPHASE  POWER 


129 


Of  all  polyphase  systems  the  two-phase  is  the  simplest.  It 
consists  of  two  single-phase  circuits  interconnected  or  not,  their 
respective  E.M.F.'s  being  displaced  90°  from  each  other.  Such 
a  system  may  be  run  four  wire  (hi  which  case  the  two  single- 
phase  circuits  may  be  entirely  independent)  or  three  wire,  in 
which  case  a  common  wire  is  used  for  the  two  single-phase  cir- 
cuits. As  this  common  wire  carries  more  current  than  the  other 
two  it  is  made  larger. 

In  case  the  phases  are  interconnected  so  as  to  give  four 
E.M.F.'s  equal  and  equally  separated  in  phase  it  is  styled  the 
quarter-phase  system.  The  connections,  for  two-phase  three 
wire,  are  given  in  Fig.  69  (a) ;  the  two  connections  for  the  quarter- 
phase  system,  star  and  mesh,  are  given  in  Fig.  69  (b)  and  (c). 


(e) 


To  study  the  E.M.F.  and  phase  relation  in  the  two-phase  cir- 
cuit make  connections  of  phases  as  given  in  Fig.  70.  Measure  the 
voltage  between  1-2  and  2-3  and  1-3.  Then  put  a  noninductive 
load  on  phase  1-3  and  connect  the  pressure  coils  of  the  watt- 
meter to  3-2.  From  the 
volt-amperes  on  the  watt- 
meter and  watts  indicated 
we  may  calculate  the  phase 
displacement  between  the 
voltage  1-3  and  3-2  (as  a 
noninductive  load  is  used, 
current  in  1-3  is  in  phase  FJG  70. 

with  voltage  1-3). 

Put  an  equal  noninductive  load  on  each  phase  and  measure 
the  current  in  each  line.  (Use  a  dynamometer  board  and  the 
same  ammeter  for  the  three  readings  so  as  to  minimize  calibra- 
tion errors  in  instruments.)  Measure  the  power  supplied  by 
each  phase,  using  one  wattmeter,  connecting  its  current  coil 
successively  in  each  of  the  two  outside  lines  and  its  potential 


130 


ALTERNATING  CURRENTS 


coil  from  each  outside  to  the  common  wire  as  in  Fig.  71  (a). 
Then  measure  the  power  with  the  same  load  by  inserting  the 
current  coil  of  the  wattmeter  in  the  common  wire  and  connecting 
its  potential  coil,  one  side  to  the  common  and  the  other  to  the 
two  outer  wires  successively,  as  in  Fig.  71  (b).  Make  the  same 
tests  with  inductive  load. 


OOOdCiL 


000000 

b 


FIG.  71. 


The  power  factor  of  the  load  may  be  found  by  the  formula: 
Watts  per  phase  =  (El)  (per  phase)  X  cos  0. 

When  the  wattmeter  is  connected  as  in  Fig.  71  (b),the  potential 
coil  being  connected  to  upper  line,  the  reading  of  the  meter  will 
give 

Wi  =  El  V2  cos  (45°  +  0), 

and  with  it  connected  to  the  lower  line  will  read 

W2  =  El  V2cos(45°  -0), 
so  that 

Wl  +  TF2  =  El  V2  (2  cos  45°cos  0)  =  2  El  cos  0. 

This  is  evidently  the  true  power  of  the  system.  It  will  be  no- 
ticed that  when  the  power  factor  of  the  system  is  not  =  1,  then  the 
two  wattmeter  readings  will  not  be  alike;  if  the  phase  angle  of  the 
load  is  45°  one  of  the  meters  will  read  zero.  If  the  lag  or  lead  is 
more  than  45°  one  of  the  meters  will  read  negatively,  under  which 
condition  its  potential  coil  (or  current  coil)  must  be  reversed, 
the  reading  taken  and  called  negative,  in  which  case  the  total 
power  supplied  =  W\  —  Wz. 

The  power  factor  of  the  system  may  be  obtained  from  watt- 
meter readings  alone  as  shown  by  the  following  derivation. 

TFi  +  W2  =  El  2  V2  (cos  45°  cos  0), 
Wi  -  Wz  =  El  2  \/2  (sin  45°  sin  0), 


so  that 


POLYPHASE  POWER  131 

Calculate  0,  for  the  inductive  load,  by  this  method  and  compare 
with  value  obtained  by  formula  (1). 

By  means  of  the  wattmeter  prove  that  the  two  single-phase 
circuits  do  have  their  E.M.F.'s  differing  in  phase  by  90°. 

Make  the  two  connections  for  quarter-phase  system  and  meas- 
ure the  different  voltages.  If  one  of  the  coils  is  connected  wrong 
in  the  star  connection  (i.e.,  180°  out  of  phase),  the  different 
quarter-phase  voltages  will  not  be  equal.  In  making  the  mesh 
connection,  before  the  mesh  is  closed,  measure  the  total  E.M.F. 
in  the  mesh  by  connecting  a  voltmeter  across  the  open  "  corner." 
This  should  be  zero  but  will  not  be  in  case  one  of  the  coils  is 
reversed.  If  the  mesh  is  closed  with  one  of  the  coils  wrongly 
connected  a  large  current  will  flow  around  the  closed  circuit.  In 
each  of  the  quarter-phase  connections  measure  the  diametrical 
voltage  and  each  quarter-phase  voltage. 

Plot  all  E.M.F.  and  current  measurements  as  vectors  on  sec- 
tion paper  and  explain  the  different  results  obtained. 


EXPERIMENT   XXV. 

CURRENT  AND  E.M.F.  RELATIONS  IN  A  THREE-PHASE  CIRCUIT; 

POWER    AND    POWER    FACTOR;  "EQUIVALENT" 

RESISTANCE  AND  CURRENT. 

A  THREE-PHASE  system  is  one  in  which  the  voltages  generated 
in  the  three-phase  windings,  from  which  the  three-phase  system 
is  obtained,  are  120°  apart.  That  this  angular  displacement  of 
E.M.F.'s  is  so  follows  from  the  placing  of  the  coils  on  the  arma- 
ture. These  three  single-phase  windings  may  be  connected  to- 
gether in  star  (or  Y)  shown  in  Fig.  72  (a) ;  or  they  may  be  connected 
in  delta  (or  mesh)  shown  in  Fig.  72  (b).  When  connected  in  delta 
the  line  voltage  is  quite  evidently  the  voltage  generated  per 


FIG.  72. 

phase.  In  the  Y  connection  this  is  not  so  and  the  relation 
of  line  voltage  to  phase  voltage  constitutes  the  first  part  of  this 
test.  The  three-phase  voltages  are  represented  by  equal  vectors 
spaced  120°. 

If  equal  noninductive  loads  are  connected  to  each  of  the  three 
phases  it  is  evident  that  the  three  lines  I/,  2'  and  3',  in  Fig.  73, 
will  be  carrying  equal  currents  120°  out  of  phase  with  each  other. 
But  the  sum  of  three  equal  vectors  120°  apart  is  zero.  Hence 
the  three  lines  1',  2'  and  3'  may  be  joined  together  for  their 
whole  length  and  no  current  will  flow  in  the  resulting  conductor. 
But  if  no  current  flows  in  it  the  conductor  is  useless  and  so  is  not 
used  and  we  have  the  normal  three-wire  Y  connection  given  in 
Fig.  72  (a).  The  current  in  line  1  will  be  in  phase  with  the 
E.M.F.  of  phase  1  —  1',  that  in  line  2  will  be  in  phase  with  E.M.F. 
in  phase  2  —  2',  etc. 

132 


POLYPHASE  POWER 


133 


Now  to  get  the  voltage  between  lines  b  and  c,  Fig.  73,  it  is 
evident  that  the  voltage  Ob  must  be  subtracted  vectorially  from 
the  voltage  Oc.  But  vector  subtraction  is  performed  by  revers- 
ing the  vector  to  be  subtracted,  then  performing  vector  addition. 
This  is  shown  on  the  vector  diagram,  Fig.  73,  as  Ob'  and  the 
voltage  between  lines  b  and  c  will  be  the  resultant  of  Ob'  and  Oc, 
given  in  the  diagram  by  OB.  Now  the  angle  c  OB  =  BOb'  =  30°, 
hence  OB  =  Oc  Vs.  In  the  same  way  the  voltage  between  lines 
c  and  a  is  given  by  the  vector  OC  and  between  lines  a  and  6 


FIG.  73. 


is  given  by  the  vector  OA.  We  have,  therefore,  the  fact  that 
in  the  Y  connection  the  line  voltages  are  120°  apart  and  equal 
to  V3  X  (phase  voltage) .  The  line  current  is  evidently  the 
same  as  the  phase  current. 

In  the  delta  connection  the  relation  between  phase  current 
and  line  current  may  be  also  determined  by  vector  diagram. 

Suppose  the  phases  are  not  connected,  but  each  is  supply- 
ing a  noninductive  load,  Fig.  74.  Then  the  currents  in  the 
different  circuits  will  be  120°  apart  as  represented  by  the  vec- 
tors, Fig.  74  (b).  Now  if  the  two  lines  1  and  1'  be  replaced 
by  a  single  line  the  current  in  this  line  would  evidently  be  current 
in  (a)  minus  current  in  (b) .  In  the  vector  diagram  this  is  shown 
by  reversing  vector  Ob  to  Ob',  then  adding  Oa  and  Ob'  to  give  the 
line  current  OA.  In  the  same  way  when  the  lines  2,  2'  are  joined 
and  3,  3'  their  respective  currents  will  be  given  in  the  vector 
diagram  by  OC  and  OB,  respectively.  In  the  delta  connection 
then  the  line  currents  are  120°  apart  and  equal  to  the  (phase 
current)  X  V3.  (Proof  same  as  for  E.M.F.'s  in  Y  connection.) 


134 


ALTERNATING  CURRENTS 


For  either  connection  it  is  evident  that  if  e  and  i  represent  the 
phase  voltage  and  current  respectively,  when  the  loads  are 
balanced,  the  power  being  supplied  by  the  three-phase  system 


(6) 


FIG.  74. 


=  3ei  (suppose  p.f.  =  1)  or  =  3  ei  cos  </>  if  the  load  is  inductive. 
Calling  E  and  /  the  line  E.M.F.  and  current  we  have 


in  star  connection      E  =  e 
in  delta  connection     E  =  e, 


/  =  i. 
I  =  i  V3. 


1 

'2-1 

^e                    j^ 

<30\                                                              >X^ 

NV                                                           ^S-2 

. 

^3           \ 

*^M 

(a)                                                    (I 

FIG.  75. 

Using  these  values  in  above  expression  for  power,  we  have 
Power  supplied  by  three-phase  system  =  EI  V%  cos  </>, 

and  this  expression  is  the  same  whether  star  or  delta  connections 
are  used. 

These   relations  between  E.M.F.  and  current  may  now  be 
collected  and  we  may  represent  vectorially  the  currents  in  the 


POLYPHASE   POWER 


135 


lines  and  the  E.M.F.'s  between  lines  for  each  connection,  first 
considering  noninductive  load.  Conditions  for  Y  connections 
are  given  in  Fig.  75  (a)  and  for  delta  connection  by  Fig.  75  (b) . 
If  the  load  is  inductive  of  power  factor  =  cos  0,  the  conditions 
are  represented  for  the  two  cases  by  Fig.  76. 


E 


t-i 


FIG.  76. 


Now,  if  it  is  desired  to  measure  the  power  output  of  the  three- 
phase   system,  only  two  wattmeters   are  necessary  as   will  be 
shown.     Suppose  the  current  coils  of  the  two  meters  are  con- 
nected in  the  lines  1  and  2.    The 
potential   coils   are   to  be   con- 
nected, one  end  to  the  line  in 
which  the  current  coil  is  placed 
and  the  two  free  ends  both  con- 
nected to  the  third  line.     The 
meters  will  be  as  represented  in 
Fig.  77.     The  reading  of  watt- 
meter No.  1  will  be  I\  E2i  cos 
(0  -  30°)  and  wattmeter  No.  2 
will  read  73  E23  cos  (0+30°).    In  the  balanced  system 
and  E 12  =  E^  =  #31,  so  that 


FIG.  77. 


/2=/3 


=  EI{  cos  (0  +  30°)  +  cos  (0  -  30°)  \  =  2  El  cos  0  cos  30 
=  El  V3  cos  0. 


But,   as  has   been   shown,    El  V3  cos   0   is   the    power   de- 
livered by  the  three-phase  system,  hence  the  sum  of  the  readings 


136  ALTERNATING  CURRENTS 

of  two  wattmeters  so  connected  will  give  the  total  power  of  the 
three-phase  circuit.* 

From  inspection  of  the  two  equations  just  given  it  is  evident 

that  the  power  factor  of  three-phase  circuits  =  —  -  —  -/=^-' 

If  0  =  60°,  then  W2  =  El  cos  90°  =  0  and  if  0  is  greater  than 
60°  the  meter  W2  will  be  deflected  backwards.  Its  potential  coil 
must  then  be  reversed  and  the  reading  be  called  negative.  If  0 
is  such  that  the  current  leads  (instead  of  lags)  then  Wi  reads 
zero  at  0  =  60°  and  reverses  for  greater  values  of  0. 

For  values  of  0  less  than  60°,  power  =  Wi  +  W2  and  for  0 
greater  than  60°,  power  =  (Wi  —  Wz).  It  is  quite  evident  that 
if  one  of  the  meters  is  indicating  negative  power  it  must  be 
the  one  having  the  smaller  reading.  When  doubt  exists  as  to 
whether  or  not  the  power  factor  of  the  circuit  is  less  than  0.5 
(i.e.,  0  >  60°)  the  power  factor  of  the  load  should  be  increased 
somewhat;  if  the  indication  of  one  of  the  meters  decreases,  it  is 
recording  negatively. 

All  of  the  above  discussion  supposes  a  balanced  load.  It  can 
be  shown  that  two  wattmeters  will  measure  the  power  of  the 
three-phase  circuit  for  any  condition  of  balance  and  power 
factor.  In  case  the  load  is  unbalanced  the  term  "  power  factor  " 
loses  its  significance,  as  it  may  be  figured  out  to  have  several 
different  values  according  to  how  the  readings  of  E  and  I  are 
used  and  averaged. 

If  it  is  known  that  the  load  is  balanced,  then  only  one  watt- 
meter is  necessary.  The  current  coil  is  connected  in  one  line 
and  one  end  of  the  potential  coil  to  the  same  line.  A  reading  is 
taken  with  the  free  end  connected  to  each  of  the  other  lines  and 
the  sum  of  the  two  readings  gives  total  power. 

*  Another  analysis  which  shows  that  the  two  wattmeters  do  record  all  of 
the  three-phase  power  is  as  follows  : 

Let  ii,  iz,  iz,  be  the  instantaneous  values  of  the  currents  in  the  three  lines. 
Let  ei,  e2,  e3,  be  the  instantaneous  values  of  voltage  across  the  three  phases  of 

the  load,  supposing  a  Y-connected  load. 
Then  we  have 


Wi  +  Wz  =  eiii  +  e2iz  +  e3  (it  +  ii). 

But,  evidently,  at  any  instant  we  must  have  ii  +  it  =  i$  so  that  Wi  +  Wz  = 
e\i\  +  e&z  +  e3i3,  which  is  the  total  three-phase  power,  at  any  instant.  Hence 
the  two  wattmeters  record  at  any  instant  the  total  three-phase  power,  irrespec- 
tive of  power  factor  or  balance. 


POLYPHASE  POWER  137 

The  power  factor  of  the  three-phase  circuit  may  be  measured 
with  the  use  of  ammeter,  voltmeter  and  wattmeter;  or  from  the 
equations  derived  for  the  readings  of  W\  and  TF2  it  is  seen  that 


Tan0  =  (derive  this  formula) 

W  i  T   rr  2 

so  that  ammeter  and  voltmeter  are  not  necessary  to  find  $. 

Polyphase  circuits  are  more  readily  solvable  when  their 
quantities  are  expressed  in  "  equivalent  "  single-phase  quanti- 
ties. "  Equivalent  "  single-phase  current  is  the  value  of  /, 
which,  multiplied  by  E  cos  </>,  gives  total  power  of  circuit.  In 
three-phase  circuit  equivalent  single-phase  current  /'  =  V3  X  /. 
The  equivalent  single-phase  resistance  is  that  quantity,  which, 
multiplied  by  (I')2,  gives  total  energy  used  as  heat  in  the  system. 
For  either  a  Y  or  delta  load,  Rr  is  equal  to  one-half  the  re- 
sistance measured  between  lines. 

With  lamp  banks  for  load  prove  the  relation  of  voltage  and 
current  in  line  and  phase  for  Y  and  delta  connections  of  load. 
Also  by  two  wattmeters  measure  the  power  of  the  three-phase 
system  and  prove  (by  measuring  the  watts  in  the  separate  phases) 
that  Wi  +  W2  =  total  power,  both  for  balanced  load  and  un- 
balanced load. 

Obtain  a  three-phase  load  (Y  connection  is  the  simpler)  whose 
power  factor  can  be  made  small.  Take  readings  of  Wi  and  W2 
(also  of  watts  per  phase)  with  <£  less  than  60°  andjgreater  than 
60°  for  balanced  and  unbalanced  load.  Account  for  values 
observed.  Calculate  the  equivalent  resistance  and  current  for 
the  inductive  load. 

To  balance  a  Y-connected  load,  open  phase  3,  and,  by  adjust- 
ing for  equal  voltage  drop  across  phases  1  and  2,  balance  these 
two.  Then  open  phase  1  and  balance  phases  2  and  3,  making  all 
adjustment  on  phase  3;  then  phase  1  may  be  connected  and  the  load 
will  be  balanced. 


EXPERIMENT  XXVI. 

GENERAL    POLYPHASE    TRANSFORMATION;    TWO-PHASE    TO 

THREE-PHASE  TRANSFORMATION  WITH  BALANCED 

AND  UNBALANCED  LOAD. 

IT  is  impossible  to  change,  by  means  of  static  transformers, 
single-phase  power  into  polyphase  power,  but  having  one  poly- 
phase system,  it  is  possible,  by  selecting  transformers  of  proper 
ratio  and  properly  connecting  them,  to  change  to  any  other  poly- 
phase system.  In  a  balanced  polyphase  system  whether  two-, 
three-  or  six-phase,  the  power  supply  is  uniform  and  constant, 
whereas,  in  a  single-phase  system,  it  pulsates  between  zero  and 
a  maximum.  Now,  aside  from  the  small  amount  of  energy 
stored  in  its  magnetic  field,  it  is  quite  evident  that  a  static  trans- 
former cannot  act  as  a  reservoir  for  energy,  i.e.,  cannot  take  it 
in  at  one  instant  and  give  it  out  the  next.  Hence  it  is  evident, 
from  the  standpoint  of  energy,  that  single-phase  pulsating  power 
cannot  be  transformed  by  ordinary  static  devices  into  a  poly- 
phase system  giving  off  a  constant  supply  of  power. 

Single-phase  power  can,  however,  be  changed  into  a  balanced 
polyphase  system  by  the  use  of  rotating  apparatus.  In  such,  the 
average  single-phase  power  input  will  equal  the  constant  value 
of  polyphase  power  output  (neglecting  the  small  losses  in  the 
machine  itself).  When  the  input  is  in  excess  of  the  output 
the  surplus  input  will  be  stored  in  the  form  of  kinetic  energy 
in  the  rotating  member,  and  will  be  given  out  again  when  the 
input  power  falls  below  the  output. 

Using  exactly  the  same  line  of  argument,  it  can  be  proved  that 
it  is  impossible  to  draw  single-phase  power  from  a  polyphase 
system  and  keep  the  polyphase  system  balanced  when  using 
static  transformers.  When  such  a  transformation  is  attempted 
it  will  be  found  that  the  polyphase  system  is  not  evenly  loaded, 
which  means  that  its  power  supply  is  pulsating;  in  fact,  the 
power  supply  will  be  just  as  variable  as  is  the  single-phase  output. 

It  has  been  remarked  that  any  polyphase  system,  in  which 
the  load  is  balanced,  supplies  constant,  nonpulsating  power  (the 
student  should  prove  this  point  both  analytically  and  geometri- 

138 


POLYPHASE  POWER 


139 


cally).  Such  being  the  case,  there  is  no  reason  from  the  stand- 
point of  energy  why  one  polyphase  system  cannot  be  changed 
by  static  transformers  into  any  other  polyphase  system.  If  one 
system  is  balanced,  its  power  supply  is  constant  so  the  other 
system  must  also  be  delivering  constant  power,  i.e.,  it  also  must 
be  balanced.  If  the  load  on  one  system  is  unbalanced  it  may  be 
imagined  as  a  balanced  polyphase  load  (which  is  delivering  con- 
stant power  and  which  must,  therefore,  be  supplied  by  constant 
power)  and  superimposed  upon  it  a  single-phase  load  which 
utilizes  pulsating  power  and  must  so  be  supplied  by  pulsating 
power.  This  component  of  the  total  load,  i.e.,  the  amount  by 
which  it  is  unbalanced,  must  be  supplied  by  one  phase  of  the 
polyphase  supply  and  so  the  supply  system  will  be  unbalanced. 

The  transformation  from  polyphase  to  single-phase  power  can 
be  accomplished  without  unbalancing  the  polyphase  system  if 
rotating  machines  are  used.  In  this  transformation  the  rotating 
member  acts  as  an  energy  reservoir  so  that  a  pulsating  power 
may  be  drawn  out  while  constant  power  is  supplied. 

It  will  be  shown  how  any  polyphase  system  can  be  transformed 
into  any  other,  with  special  reference  to  the  two-phase  three- 
phase  transformation. 

Consider  a  two-phase  supply  and  two  separate  transformers 
connected  to  the  two  phases,  the  secondaries  to  have  a  number 
of  taps  (as  shown  in  Fig.  78),  so 
that  different  ratios  may  be  obtained. 
Let  A  and  B  represent  the  two  trans- 
formers with  extra  taps  as  shown. 
The  quarter-phase  star-connected 
system  could  be  obtained  by  con- 
necting together  taps,  2,  3,  2'  and  3'. 
If  a  quarter-phase  system  of  lower 
voltage  is  desired  the  coils  may  be 
connected  in  mesh. 

To  transform  a  two-phase  to  three- 
phase  system  the  T  or  "  Scott  "  connection  is  used.  On  one  of 
the  transformers  (A  in  Fig.  79),  an  extra  tap,  5,  is  brought  out. 
The  tap  is  so  connected  that  the  ratio  of  the  voltage  between  1 
and  4  is  to  that  between  1  and  5  as  1:  0.866.  Tap  I'  is  con- 
nected to  the  junction  of  taps  2  and  3  and  the  three-phase  sys- 
tem is  obtained  from  taps  1,  4  and  5  as  shown  in  Fig.  79,  where 
a,  b,  c  is  the  three-phase  line. 


A 

3QQQOQQQQO 


FIG.  78. 


140 


ALTERNATING   CURRENTS 


That  such  a  connection  will  give  a  true  three-phase  system, 
the  line  voltages  being  equal  and  120°  out  of  phase  with  each 

other,  will  be  shown  by  the  vec- 
tor diagram  of  Fig.  80.  The  sec- 
ondary voltages  of  A  and  B  will 
be  90°  apart  but  not  equal,  as 
there  are  fewer  turns  between  I' 
and  5  than  between  1  and  4. 
If  the  voltage  1-4  is  taken  as 
100  volts  then  voltage  l'-5  = 
86.6  volts.  The  vector  diagram 
of  the  line  voltages  is  shown  in 
Fig.  80.  In  plotting  the  diagram 
of  E.M.F.  the  actual  connection 
as  given  in  Fig.  79  must  be  con- 
sidered, to  show  whether  vectors  must  be  added  or  subtracted. 
The  voltage  ab  is  100  and  is  put  in  its  proper  phase  in  Fig.  80 
directly  from  Fig.  79.  The  vol-  354 
tage  b-c  is  the  resultant  of  4-3 
and  3-5.  But  as  both  these  vol- 
tages act  toward  b,  they  must 
be  reversed  in  direction  and 
added  to  get  the  voltage  be. 
The  resultant,  4-5,  is  plotted  in 
the  lower  part  of  Fig.  80,  as  b-c. 
The  magnitude  of  voltage  b-c 
is  equal  to  V86.62  +  502  =  100 

volts.  Its  phase  position  with  respect  to  ab  is  (90°  +  tan"1  —7=] 
=  120°.  V  V3/ 

Therefore,  be  =  ab,  and  as  vectors  they  are  120°  apart,  and 
be  is  so  shown  in  dotted  lines  in  Fig.  80. 

The  voltage  ca  is  evidently  the  vector  difference  of  5—1'  and 
3-1,  or  the  vector  sum  of  5-1'  and  3-1  reversed,  and  is  so  con- 
structed in  Fig.  80.  Its  magnitude  =  V86.62  +  502  =  100  volts 
and  its  phase  angle  with  respect  to  be  is  120°,  and  it  is  so  plotted 
in  dotted  lines  in  Fig.  80.  So  that  this  "  T  "  connection  of 
transformers  changes  two-phase  to  three-phase  power. 

The  neutral  point  of  this  three-phase  system  may  be  obtained 
by  bringing  out  a  tap  in  transformer  A,  one-third  of  the  way 
from  I7  to  5  as  indicated  at  X. 


FIG.  80. 


POLYPHASE  POWER 


141 


The  question  of  power  furnished  by  each  transformer  is  im- 
portant. If  we  assume  a  noninductive  load  of  10  amperes  per 
line  on  the  three-phase  Ipe,  the  load  and  volt-ampere  readings 
of  the  two  transformers,  on  the  three-phase'  side,  will  be: 


Transformer  A 
Transformer  B 


E 

86.6 
100 


7 

10 
10 


Coa<f> 

1.00 
.866 


Watts 

866 
866 


Volt-amperes 

866 
1000 


From  this  it  is  seen  that  in  the  Scott  connection  slightly  more 
transformer  capacity  is  required  than  in  the  delta  or  star  three- 
phase  connection,  in  both  of  which  cases  the  total  volt-amperes 
for  all  transformers,  for  noninductive  load,  is  equal  to  the  watts 
load,  whereas  in  the  T  connection  the  transformers  must  have 
about  8  per  cent  greater  capacity. 

Any  vector  may  be  resolved  into  two  components  at  right 
angles  to  each  other.     Hence  any  polyphase  system  may  be 
obtained  from  a  two-phase  system,  the  only  requisite  being  that 
the  transformer  ratios  must 
be  certain  values  and  that 
the  sections  of  the  two  trans- 
formers must  be  connected 
in  series  in  right  relations  to 
one  another  (i.e.,  direct  or 
reversed) . 

With  two  transformers  ar- 
ranged for  Scott  connection, 
connect  to  a  two-phase  sup- 
ply as  in  Fig.  81.     With  no  -pio.  81. 
load  on  the  three-phase  side, 

measure  all  necessary  voltages  to  prove  that  the  above  given 
vector  construction  is  correct.  The  equality  of  the  three-phase 
voltages  may  be  directly  tested  by  the  voltmeter.  That  the 
three  voltages  must  be  120°  apart  may  be  seen  by  supposing  the 
three  lines  connected  by  equal  resistances  as  in  Fig.  82.  By  volt- 
meter we  measure  the  magnitude  of  the  voltage  AB'}  then  that  of 
BC  and  then  that  of  CA.  But  as  A  cannot  have,  at  any  instant, 
more  than  one  value  of  potential,  so  that  in  traversing  the  circuit 
ABC  we  come  back  to  a  point  of  the  same  potential  as  that  from 

Note.  —  It  is  left  for  the  student  to  prove  by  vector  construction  the  power 
factors  assumed  above  for  the  transformers  A  and  B  and  also  the  statement 
made  in  regard  to  the  A  and  Y  connection  of  transformers. 


142  ALTERNATING  CURRENTS 

which  we  started,  the  three  voltages  AB,  BC,  and  CA,  when  plotted 
as  a  triangle,  must  close.  But  the  only  triangle  having  sides  of 

A  equal  magnitude  is  one  in 

which  the  sides  are  120° 
apart.  Of  course  it  must  be 
remembered  that  if  the 
sides  of  an  equilateral  tri- 
FlG  82  angle  are  considered  as 

vectors,  plotted  in  the  same 

direction  around  the  triangle,  these  vectors  are  120°  apart  and  not 
60°  as  might  be  supposed  if  the  inside  angles  of  the  triangle  were 
considered. 

Now  put  a  balanced  noninductive  load  on  the  three-phase  side 
and  note  whether  the  two-phase  side  is  balanced.  Take  similar 
readings  with  two  conditions  of  unbalance  on  the  three-phase 
side.  By  taking  suitable  meter  readings  when  the  three-phase 
load  is  balanced,  show  that  the  power  factors  assumed  for  trans- 
formers A  and  B  in  the  foregoing  discussion  are  correct.  If  the 
three-phase  system  is  only  loaded  in  phase  ab  how  much  load 
will  each  transformer  carry?  If  only  loaded  on  phase  ac  or  be 
how  will  the  two-phase  system  be  loaded?  Prove  both  of  these 
answers  by  suitable  measurements. 


EXPERIMENT  XXVII. 

THREE-PHASE  TRANSFORMATION;  HIGHER  HARMONICS  IN 
THREE-PHASE  CIRCUITS. 

THE  transformation  of  a  three-phase  system  from  one  voltage 
to  another  may  be  accomplished  by  means  of  one  three-phase 
transformer,  also  by  two  or  three  single-phase  transformers. 

From  the  standpoint  of  first  cost  and  electrical  efficiency  the 
one  three-phase  transformer  is  to  be  preferred,  but  for  reliability 
of  operation  the  three  single-phase  transformers  give  better 
results.  The  possible  connections  of  the  single-phase  trans- 
formers are 
With  three  transformers: 

(1)  Primaries,  delta  secondaries,  delta 

(2)  Primaries,  star  secondaries,  star 

(3)  Primaries,  star  secondaries,  delta 

(4)  Primaries,  delta  secondaries,  star 
With  two  transformers : 

(5)  Primaries  T-connected          secondaries  T-connected 

(6)  Primaries  V-connected          secondaries,  V-connected 

(7)  Primaries  V-connected          secondaries,  Y-connected. 
The  different  methods  give  different  ratios  of  transformation  and 
also  different  possible  output  of  transformer  groups. 

(1)  and  (2)  give  the  same  ratio  of  transformation,  and  the 
possible  output  of  the  group  is  equal  to  the  sum  of  the  ratings 
of  the  three  transformers  because  the  power  factor  of  each 
transformer  is  one,  on  noninductive  load. 

(3)  and  (4)  also  have  a  group  rating  equal  to  the  sum  of  the 
separate  capacities.  In  (3)  the  ratio  of  transformation  (for  1 :  1 
transformers)  is  \/3: 1,  while  the  same  transformers  connected 
as  in  (4)  give  a  ratio  of  voltage  of  1 :  V3. 

For  method  (5)  the  two  transformers  are  connected  as  in  Fig.  83. 
The  ratio  of  the  transformers  may  be  1 : 1  and  not  1 :  0.866  as  in 
the  Scott  two-three-phase  transformation.  The  T  connection, 
however,  gives  a  group  capacity  only  equal  to  86.5  per  cent 
that  of  the  sum  of  the  capacities  of  the  individual  transformers. 
This  is  due  to  the  fact  brought  out  in  Experiment  26,  that  even 
when  the  three-phase  load  is  noninductive,  one  of  the  trans- 
formers furnishes  a  current  of  30°  out  of  phase  with  its  E.M.F. 

143 


144 


ALTERNATING  CURRENTS 


With  a  noninductive  load  the  three-phase  voltages  with  the 
T  connection  will  remain  equal,  but  with  inductive  load  they 
will  become  unbalanced  owing  to  the  fact  that  the  current  in 
one  leg  of  the  transformer  A  tends  to  come  into  phase  with  the 
transformer  E.M.F.  when  the  load  current  is  lagging,  and,  as 
proved  in  an  earlier  experiment,  a  transformer  regulates  better 
on  noninductive  than  on  inductive  load.  This  effect  is  dia- 
grammed in  Fig.  84,  which  gives  the  vector  relations  of  E  and 
I  in  the  two  transformers.  The  vectors  a,  6,  c  show  the  line 


FIG.  83. 


FIG.  84. 


currents  on  the  secondary  side  of  the  transformers  with  non- 
inductive  load  while  a',  b',  c'  show  the  same  quantities  when  the 
line  currents  lag  by  an  angle  <f>. 

The  half  of  transformer  A  which  is  carrying  current  a'  will 
regulate  worse  than  the  other  half  because  of  the  greater  lag 
angle  of  its  current.  It  is  to  be  noted  that  B  will  have  less  than 
rated  core  loss. 

Method  (6)  is  very  frequently  employed  where  an  early 
increase  in  power  consumption  is  expected.  If,  e.g.,  the  present 
load  is  2000  K.V.A.  and  it  is  expected  to  rise  to  3000  K. V. A.,  then 
it  is  likely  that  two  1000  K.V.A.  transformers  will  be  installed 
and  connected  in  V.  Then  when  the  capacity  is  to  be  increased 
the  third  1000  K.V.A.  transformer  will  be  added,  completing 
the  delta  installation.  With  the  two  1000  K.V.A.  transformers 
in  V  the  group  capacity  will  not  be  2000  K.W.  because  of  the 
phase  relations  of  E  and  /  in  the  transformers.  Their  combined 
capacity  will  be  about  1730  K.W.  and  adding  the  third  1000 
K.V.A.  unit  will  raise  the  group  capacity  to  3000  K.W. 

Method  (7)  does  not  give  a  balanced  three-phase  system  on 
the  secondary  and  so  is  seldom  used.  It  gives  what  is  called 
the  "  unsymmetrical  Y." 

In  connecting  up  for  method  (6)  it  is  quite  likely  that  the 


POLYPHASE  POWER 


145 


Time 


secondaries  may  be  so  connected  that  they  are  in  opposition 
(Y)  instead  of  delta  and  for  this  reason,  and  also  to  investigate 
the  vector  relations  hi  the  circuit,  method  (7)  will  be  tried. 

The  question  of  upper  harmonics  in  the  E.M.F.  wave  of  an 
alternator  becomes  of  importance  in  connecting  the  three-phases 
of  the  alternator  if  the  machine  is  delta  wound,  or  if  the  phases 
of  the  alternator  are  connected  in  Y,  then  the  upper  harmonics 
may  cause  trouble  in  connecting  transformers  to  the  line  and 
so  the  question  will  be  investigated  at  this  point. 

If  there  were  any  even  harmonics  in  the  E.M.F.  wave  of  an 
alternator  the  two  halves  of  the  wave  would  not  possess  "  mirror 
symmetry/'  i.e.,  if  the 
negative     loop     were 
moved  backward   180° 
the  two  loops  would 
not  be   symmetrical    ~ 
with  respect  to  the  time 
axis.     This  is  shown  in 
Fig.    85,    which    illus- 
trates a  wave  contain-  pjG  85 
ing  no  even  harmonics 

because  the  loop  b'  (b  moved  back  180°)  is  symmetrical  with  a, 
when  the  time  axis  is  used  as  reference  line. 

That  an  alternator  can  generate  only  such  waves  becomes 
self-evident  when  the  process  of  E.M.F.  generation  is  considered. 

If  the  winding  of  a  single- 
phase  alternator  be  devel- 
oped as  in  Fig.  86,  it  is  seen 
to  be  made  up  of  sets  of  in- 
ductors, a,  b,  c,  in  series  with 
each  other  and  these  sets  of 
inductors  are  exactly  similar. 
Hence,  whatever  E.M.F.  is 
generated  at  a  certain  instant 
will  be  exactly  duplicated  180°  later  when  inductors  b  occupy 
the  field  previously  occupied  by  c,  and  inductors  a  are  in  the  same 
field  previously  occupied  by  inductors  6.  Hence,  whatever  is 
taking  place  at  any  instant  is  exactly  duplicated  (in  opposite 
sense)  180°  later.  This  similarity  of  the  positive  and  negative 
loops  does  not  necessitate  the  loops  themselves  having  a  sym- 
metrical form  with  respect  to  an  axis,  through  the  point  of 


FIG.  86, 


146  ALTERNATING  CURRENTS 

maximum   value,   and,   in   general,  they   do   not    possess  this 
symmetry. 

It  follows  that  there  are  no  even  harmonics  in  an  E.M.F.  wave 
but  there  are  nearly  always  some  odd  harmonics,  those  of  the 
lower  frequencies  having  the  larger  amplitude  unless  some 
special  condition  emphasizes  one  of  the  higher  ones.  If  the 
magnetic  field  was  constantly  symmetrical,  and  of  proper  distri- 
bution, with  respect  to  the  centers  of  the  poles  then  no  odd  har- 
monics would  appear,  but  armature  reaction,  inequalities  in  the 
air  gap,  etc.,  act  to  prevent  this. 

The  above  remarks  on  generation  of  upper  harmonics  have 
been  made  upon  the  assumption  that  the  strength  of  magnetic 
field  was  constant  with  respect  to  time,  i.e.,  that  the  field  was 
possibly  distorted,  being  stronger  in  one  part  of  the  pole  face 
than  in  another,  but  that  it  did  not  pulsate.  If,  however,  the 
field  is  not  constant  then  odd  harmonics  will  be  produced  by  this 
variation  of  field  strength.  In  an  alternator,  having  few  slots 
per  pole,  the  reluctance  of  the  magnetic  circuit  changes  con- 
siderably with  the  angular 
position  of  the  armature  be- 
cause at  certain  times  there 
will  be  one  more  tooth  under 
the  pole  face  than  at  others. 
This  will  be  understood  by 
reference  to  Fig.  87,  which 
supposes  a  three-phase  arm- 
ature having  one  coil  per 
phase  per  pole.  Such  an 
armature  will  cause  pulsa- 
tions in  the  field  of  six  times  the  frequency  of  the  armature 
E.M.F.  Calling  the  frequency  of  these  field  changes  p,  and 
that  of  the  generated  E.M.F.  in  the  armature  co,  then  there  will 
be  superimposed  in  the  frequency  co  a  harmonic  of  frequency 
(p  —  co)  which  will  always  be  odd,  because  p  is  an  even  multiple 
of  co.*  In  a  certain  machine  having  a  three-phase  winding  of 
two  coils  per  pole  per  phase,  the  air  gap  being  small,  there  is  a 
very  pronounced  eleventh  harmonic. f 

*  A  more  complete  explanation  of  the  effect  of  field  variations  upon  wave 
shapes  in  the  rotating  armature  will  be  given  shortly.  Prof.  Pupin  has  been 
doing  a  deal  of  experimental  work  on  the  question  and  the  results  will  be 
published  soon. 

f  For  curve  of  E.M.F.  generated  by  this  machine  see  Plate  7  of  the  Appendix. 


POLYPHASE   POWER  147 

The  distorted  curve  showing  the  magnetizing  current  of  a 
transformer  has  for  its  principle  harmonic  the  third.  Un- 
doubtedly all  the  higher  odd  ones  are  present  but  only  the  third, 
fifth  and  possibly  the  seventh  will  have  appreciable  magnitude. 
If  two  harmonic  functions  are  represented  by  two  vectors  120° 
apart  it  is  evident  that  the  vectors  showing  the  third  harmonic 
of  these  functions  will  be  directly  in  phase  with  one  another. 
Hence,  if  three  transformers  are  connected  in  delta  to  a  three- 
phase  line  on  which  there  is  a  marked  third  harmonic  in  the  wave 
of  E.M.F.  their  third  harmonic  currents  will  be  in  phase  with  each 
other  and  so  will  flow  in  the  local  path  formed  by  the  primaries. 
This  third  harmonic  will  not  appear  in  the  line  but  will  do  harm 
in  heating  the  transformers.* 

If  /  =  line  current,  It  =  current  in  transformer  coil  and  73  = 
amplitude  of  third  harmonic,  then 


This  will  only  represent  approximately  the  condition  as  Is  will 
include  the  9th,  15th,  etc.,  harmonics. 

If  the  primaries  of  the  transformers  are  connected  in  Y  and 
the  secondaries  neither  connected  in  A  nor  loaded,  then  the  volt- 
age wave  form  across  a  primary  coil  will  be  much  distorted. 
The  resulting  voltage  across  any  transformer  Et  will  be  greater 

than  (-/=)  X  line  voltage  by  an  amount  which  equals  vectorially 
the  reaction  E.M.F.  of  the  third  harmonic. 

ET 

Expressed  algebraically  —7=  —  VEt2  —  E32.     This  condition  re- 

sults in  what  is  sometimes  called  the  "  wabbling  neutral";  the 
voltage  from  one  corner  of  the  Y  to  the  center  is  constantly 
changing  due  to  the  third  harmonic  E.M.F. 

If  a  third  harmonic  E.M.F.  is  generated  in  the  alternator  coils 
and  the  coils  are  connected  in  delta,  then  a  considerable  current 
may  circulate  in  the  armature  even  when  the  load  is  zero.  This 
will  cause  useless  heating  of  armature.  With  delta-connected 
armature,  the  third  harmonic  cannot  appear  in  the  line  E.M.F. 
(Why?)  But  in  a  star-connected  alternator  the  line  E.M.F.  will 
contain  the  third  harmonic.  If,  then,  three  transformers  are 

*  For  curves  showing  voltage  and  current  forms  in  three-phase  groups  of 
transformers  see  Appendix,  Plates  9-14. 


148  ALTERNATING  CURRENTS 

connected  in  delta  to  the  line  a  circulating  current  will  flow  in 
their  primary  coils.  In  case  the  primaries  are  connected  star  and 
the  secondaries  delta,  the  circulating  current  will  appear  in  their 
secondaries.  In  case  everything  is  connected  in  star  no  third- 
harmonic  current  will  flow  unless  the  neutral  points  of  alternator 
and  apparatus  are  grounded.* 

Make  connections  and  measure  E.M.F.'s  for  all  methods  of 
transformation.  Obtain  sufficient  readings  that  vector  dia- 
grams of  the  E.M.F.'s  may  be  constructed  for  each  connection. 
By  loading  with  balanced,  three-phase,  noninductive  load  and 
adjusting  load  until  each  transformer  is  operating  at  rated 
current,  prove  that  methods  3  and  4  have  group  capacity  equal 
to  sum  of  separate  capacities  while  methods  5  and  6  give  only 
56f  per  cent  the  capacity  of  a  group  of  three  transformers,  each 
of  same  capacity  as  one  of  the  two  used.  With  the  primaries 
of  three  equal  transformers  in  delta  and  not  loaded  get  sufficient 
data  to  calculate  the  value  of  the  third-harmonic  current  cir- 
culating in  the  delta  due  to  distorted  magnetization  curve. 

By  means  of  the  ondograph  get  the  curve  of  phase  E.M.F.  of 
a  three-phase  alternator  giving  a  distorted  wave  form.  Connect 
the  phases  in  delta  and  measure  the  circulating  current  by  an 
ammeter.  Get  the  ondograph  curve  of  this  circulating  current, 
and  on  same  sheet  get  curve  of  phase  E.M.F.  for  reference.  Get 
the  curve  of  line  E.M.F.  Connect  the  phases  in  star  and  again 
get  the  curve  of  line  E.M.F. 

Before  making  the  last  connection  for  a  delta  circuit,  whether 
alternator  or  transformer,  always  measure  the  voltage  across  the 
open  terminals.  The  last  connection  must  not  be  made  until  this 
voltage  is  practically  zero.  If  it  is  not  approximately  zero,  one 
of  the  phases  is  connected  in  the  circuit  180°  out  of  its  proper 
phase  and  it  must  be  reversed. 

In  case  the  transformers  are  connected  Y  —  A  the  open  cor- 
ner of  the  connection  may  show  quite  a  large  value  of  voltage 
due  to  the  adding  up  of  the  third-harmonic  E.M.F.'s  around 
the  A.  This  third-harmonic  voltage  will  produce  practically  no 
current  in  the  A  when  it  is  closed.  An  unbalanced  voltage 
(third  harmonic)  of  65  volts  on  some  3  K.W.,  110-220-volt  trans- 
formers caused  a  circulating  current  of  less  than  1  ampere 
when  the  A  was  closed.  As  soon  as  the  A  is  closed  a  small 

*  For  more  complete  analysis  of  the  third  harmonic  in  three-phase  circuits 
see  Franklin's  "  Electric  Waves." 


POLYPHASE  POWER  149 

third-harmonic  current  flows  in  the  secondary  circuit  and  the 
effect  of  this  current  is  to  damp  out  the  third-harmonic  E.M.F. 
in  the  primary  circuit.  This  point  may  be  investigated  by 
connecting  one  voltmeter  across  the  line  and  one  across  one 
primary  and  noting  the  readings  of  the  two  instruments  before 
and  after  the  secondary  A  is  closed.  Before  it  is  closed  the 
readings  of  the  two  voltmeters  indicate  perhaps  a  large  third- 
harmonic  E.M.F. ;  after  closing  the  secondary  this  disappears. 

Construct  vector  diagram  of  all  E.M.F.  relations  obtained  and 
explain  group  capacity,  etc.  Explain  curves  obtained  from  the 
alternator. 

Why  is  a  three-phase  transformer  more  efficient  electrically 
than  three  single-phase  transformers  of  same  combined  capacity? 
Why  are  three  single-phase  transformers  more  economical  from 
the  standpoint  of  reliability  and  maintenance  cost? 


EXPERIMENT  XXVIII. 

PHASE  CHARACTERISTICS  OF  A  SYNCHRONOUS  MOTOR;  CAPAC- 
ITY ACTION  ON  AN  INDUCTIVE  LINE,   "  HUNTING  "   OF 
SYNCHRONOUS  MACHINES. 

IF  two  alternators  are  operating  in  parallel  and  the  driving 
power  is  taken  away  from  one  of  them  it  will  continue  to  run, 
operating  as  a  motor  and  drawing  its  necessary  power  from  the 
other  alternator.  Moreover,  its  speed  will  be  exactly  the  same 
as  it  had  for  parallel  operation  as  a  generator.  From  this  fact 
it  has  derived  its  name;  a  synchronous  motor,  being  supplied 
with  power  of  a  certain  frequency,  will  always  operate  as  a  motor 
at  that  speed  which  it  would  require  as  a  generator  to  give  the 
frequency  with  which  it  is  supplied. 

A  synchronous  motor  maybe  either  single  phase  or  polyphase, 
and  to  make  the  discussion  more  simple  a  single-phase  motor 
will  be  considered.  The  characteristic  feature  of  the  synchro- 
nous motor,  outside  of  its  constant  speed,  is  the  fact  that  it  can- 
not be  started  by  supplying  power  to  its  armature  through  a 
rheostat,  as  may  be  done  with  other  types  of  A.C.  motors  and 
also  D.C.  motors.  (This  remark  will  be  qualified  later  when 
applied  to  polyphase  motors.)  The  reason  for  this  lack  of 
starting  torque  becomes  evident  when  the  relation  of  the  arma- 
ture current  and  the  magnetic  field  is  considered. 

The  field,  being  supplied  from  some  D.C.  source,  is  uniform  in 
direction  and  magnitude.  The  current  circulating  through  the 
armature  conductors  is  alternating  with  the  frequency  of  the 
supply  circuit.  It  is  a  fundamental  principle  that  a  conductor 
carrying  current,  placed  in  a  magnetic  field  (not  parallel  to  the 
field) ,  will  be  acted  upon  by  a  force  which  will  tend  to  move  the 
conductor  at  right  angles  to  itself  and  the  flux.  The  direction 
of  this  force  depends  upon  the  direction  of  the  current  and  of 
the  field;  if  either  of  them  is  reversed  the  direction  of  the  force 
reverses,  but  if  both  of  them  reverse  the  force  on  the  conductor 
will  be  in  its  original  direction. 

Now  consider  the  stationary  armature  of  a  single-phase 

150 


THE  SYNCHRONOUS   MOTOR  151 

synchronous  motor  as  in  Fig.  88.  One  coil  of  only  two  con- 
ductors A  and  B  is  considered,  as  all  of  the  winding  will  act  in 
the  same  way  as  one  coil.  At  the  instant  considered  A  is  under 
a  N  pole  and  current  is  out  of  the  paper  so  that  the  force  will 
tend  to  move  the  armature 
in  a  clockwise  direction.  In 
conductor  B  the  current  is 
in  the  opposite  direction  as 
is  also  the  field  flux  (into 
the  armature  under  N  and 
out  of  it  under  S)  so  that 
the  force  on  B  will  also  tend 
to  move  the  armature  in  the 
clockwise  direction;  hence, 
it  will  begin  to  turn.  Owing 
to  the  inertia  of  the  arma- 
ture  it  will  not  have  moved 

very  much  before  the  current  in  the  conductors  will  have  re- 
versed. The  conductors  will  still  be  in  practically  the  same 
position  as  shown  in  Fig.  88,  because  if  a  frequency  of  60,  e.g., 
is  used,  it  is  quite  evident  that  the  armature  starting  from  rest 
could  not  have  moved  far  in  T^  of  a  second;  but  if  the  current 
in  the  conductors  reverses  and  they  remain  in  a  field  of  same 
direction  as  before  then  their  force  will  be  reversed  and  the 
torque  will  be  counterclockwise.  So  that  the  motor  could  not 
start,  per  se,  from  standstill  because  the  torque  varies  in  direc- 
tion with  the  same  frequency  as  the  power  supply. 

But  if  the  armature  is  revolving  rapidly  in  a  clockwise  direction 
at  the  first  instant  considered,  then  by  the  time  the  current  re- 
verses in  direction  conductor  A  will  have  moved  under  the  S  pole 
and  B  will  be  under  the  next  N  pole.  So  in  this  case  the  current  in 
the  conductor  and  the  direction  of  the  magnetic  field  in  which  the 
conductor  is  lying  both  change  at  the  same  time  and  hence  the 
torque  will  remain  in  the  clockwise  direction.  If  A  moves  from 
a  N  pole  to  a  S  pole  in  the  time  of  one  alternation  of  the  current 
then  the  armature  is  revolving  at  synchronous  speed.  This 
discussion  gives  the  reason  why  a  synchronous  motor,  running 
at  synchronous  speed,  exerts  torque  constantly  in  one  direc- 
tion, and  if  not  at  synchronous  speed  gives  a  torque  alternating 
in  direction,  and  so  is  not  capable  of  doing  work.  So  far  as 
torque  is  concerned  a  three-phase  motor  may  be  considered  as 


152  ALTERNATING  CURRENTS 

three  single-phase  motors  with  this  exception,  that  a  polyphase 
synchronous  gives  practically  uniform  torque  while  in  a  single- 
phase  motor  it  varies  between  zero  and  a  maximum  with  twice 
the  frequency  of  the  power  supply. 

In  Experiment  15  the  conditions  to  be  obtained,  before  an 
incoming  alternator  can  be  connected  in  parallel  with  the  line, 
are  enumerated.  The  synchronous  motor  must  be  brought  to 
synchronous  speed  and  these  same  conditions  obtained  before  it 
can  be  connected  to  the  supply  line.  With  some  polyphase 
synchronous  motors,  however,  no  extra  driver  is  required  to 
bring  them  to  synchronous  speed.  A  set  of  transformers  with 
low  voltage  taps  supplies  to  the  stationary  armature  a  polyphase 
voltage  of  such  a  magnitude  that  the  armature  current  is  not 
excessive  (as  there  is  no  C.E.M.F.  with  the  armature  stationary, 
normal  voltage  would  cause  .abnormally  large  current  to  flow  in 
armature).  The  polyphase  currents  distributed  in  the  polyphase 
winding  of  the  armature  produce  a  rotating  field  (to  be  taken  up 
later  pn  in  connection  with  the  mduction  motor).  This  field 
produces  eddy  currents  in  the  pole  faces  which  react  upon  the 
armature  so  that  it  is  dragged  around  in  the  opposite  direction 
to  that  of  its  rotating  field.  It  will  accelerate,  therefore,  until 
nearly  synchronous  speed  is  reached. 

During  this  accelerating  period  there  is  no  direct-current 
excitation  in  the  field  and  the  field  circuit  is  split  up  into  sections 
to  keep  down  to  a  safe  limit  the  induced  voltage  in  the  field 
windings.  This  voltage  may  get  high  enough  to  puncture  the 
field  insulation  unless  proper  caution  is  exercised  in  employing 
this  method  of  starting.  When  the  armature  has  accelerated  up 
to  nearly  synchronous  speed,  normal  voltage  is  applied  to  the  arma- 
ture (generally  by  means  of  a  double-throw  switch  connected  to  the 
half  voltage  taps  on  one  side  and  full  voltage  on  the  other)  and  the 
field  is  gradually  excited.  The  motor  will  then  generally  pull  into 
synchronism  and  proper  phase  or  180°  out  of  phase;  in  the  latter 
case  it  must  be  forced  to  slip  one  pole  or  the  field  current  may  be 
reversed.  In  other  methods  of  synchronizing,  employing  auxil- 
iary power  for  starting,  a  lamp  or  synchronoscope  is  used;  the 
operation  of  synchronizing  may  be  done  automatically  by  a 
synchronizer. 

As  the  field  strength  and  speed  of  this  type  of  motor  are  con- 
stant the  C.E.M.F.  must  be  constant,  so  that  it  is  not  at  once  evi- 
dent how  the  motor  can  accommodate  itself  to  different  loads.  In 


THE  SYNCHRONOUS   MOTOR 


153 


D.C.  motors  the  C.E.M.F.  drops  just  enough  below  the  impressed 
E.M.F.  to  permit  the  required  current  to  flow.  In  the  synchro- 
nous motor,  the  C.E.M.F.  may  be  very  much  less  than,  equal  to, 
or  even  greater  than  the  impressed  E.M.F.,  and  still  the  motor 
will  operate  well  for  different  loads.  The  explanation  of  this 
fact  is  to  be  seen  by  means  of  the  vector  diagram  of  the  motor. 
Under  normal  operation  the  C.E.M.F.  will  be  nearly  180°  dis- 
placed from  the  phase  of  the  impressed  E.M.F.  as  shown  in  Fig. 
89,  where  OEi  =  impressed  vol- 
tage, OEm  =  counter  E.M.F.  of  ^ 
motor  and  OEr  =  their  resultant. 
If  the  impedance  of  the  arma- 
ture is  Z  then  the  current  in 

the  armature  =  — ~  and  will  lag 

Li 

nearly  90°  behind  OEr  because  of 

the  inductive  character  of  the  armature  circuit.     The  output  of 

the  motor  =  OEm  X  /  cos  a  =  OEm  X  OA. 

If  now  OEm  remains  constant  and  more  load  is  put  on  the 

synchronous  motor,  the  armature  will  be  retarded  slightly  so 

that  its  space  phase  with  respect  to  the  time  phase  of  the  im- 
pressed E.M.F.  is  different.  It 
will  still  be  running  at  synchro- 
nous speed,  however.  This  phase 
displacement  of  the  armature  re- 
sults in  E.M.F.  relations,  as  shown 
in  Fig.  90.  Here  the  resultant 
E.M.F.,  OEr,  is  much  larger  and 
hence  the  current  01  is  much 
larger.  6  has  its  same  value  but  a 
is  slightly  less.  The  motor  output 

is  again  OEm  X  01  cos  a  —  OEm  X  OA,  which  is  about  twice  as 

large  as  its  previous  value. 

If  now  the  load  be  maintained  constant  and  the  field  current 

be  varied  (thus  changing  OEm)  01  must  so  change  that  OEm  X 

01  cos  a  remains  constant. 

In  Fig.  91  are  given  the  different  conditions  under  which  this 

might  occur.     Using  the  phase  of  the  current  as  reference  line, 

and  assuming  a  certain  value  of  current  O/i,  then  the  position 

and  magnitude  of  OEr  are  immediately  obtained.     A  circle  is 

described  about  0  with  radius  equal  to  line  voltage  OE\.     If  the 


154 


ALTERNATING  CURRENTS 


motor  output  =K,  then  we  must  have  OEm  X  01  cos  a  =  K. 
Draw  a  line,  BC,  perpendicular  to  01  such  that  OD  X  01  =  K. 
Then  whatever  value  OEm  may  have  its  projection  upon  01  must 
be  equal  to  OD,  i.e.,  we  have  to  construct  a  parallelogram  hav- 
ing OEr  for  diagonal,  one  side  terminating  on  the  arc  Ei'Ei  and 
the  other  on  the  line  BC.  Two  such  constructions  are  possible, 
as  shown,  so  that  either  of  two  excitations  will  satisfy  the  condi- 
tion. For  another  value  of  current  and  the  same  load  construct 
the  line  B'C'  perpendicular  to  01  and  so  that  OD'  X  01'  =K, 
and  so  two  more  excitations  will  be  found.  In  such  a  manner 


FIG.  91. 

it  will  be  found  that  for  any  given  load  on  the  motor  (within  its 
rating)  the  armature  current  may  vary  throughout  a  wide  range, 
there  being  two  values  of  excitation  for  each  value  of  armature 
current,  the  greater  excitation  giving  a  current  leading  the 
impressed  force  EI  and  the  other  a  lagging  current.  For  any 
load  there  will  be  a  minimum  value  of  current  which  will  be  a 
singly  valued  point. 

The  curve  showing  the  relation  between  armature  current  and 
excitation  for  any  given  load  is  called  the  "  phase  characteris- 
tic "  or  "  V  "  curve  of  the  motor.  For  any  given  load  the  most 
efficient  operation  will  be  obtained  by  adjusting  the  field  excita- 
tion to  give  the  minimum  value  of  armature  current  for  that 
load,  because  under  this  condition  the  armature  PR  loss  will 
be  a  minimum.  If  other  than  this  excitation  is  used  the  PR 
loss  will  be  greater  and  also  the  core  loss  will  be  different.  For 
superexcitation  the  core  loss  increases  and  under  excitation  it 
decreases.  For  most  efficient  operation  the  total  losses  (PR  + 


THE  SYNCHRONOUS  MOTOR  155 

core  losses)  should  be  a  minimum.  This  will  always  occur  when 
the  motor  is  operating  close  to  the  bottom  of  one  of  the  V  curves. 

It  will  be  noticed  in  Fig.  91  that  overexciting  the  motor  makes 
the  current  lead  the  impressed  E.M.F.,  i.e.,  the  overexcited  syn- 
chronous motor  acts  like  a  condenser,  the  amount  of  condenser 
action  depending  upon  the  amount  of  superexcitation. 

As  the  ordinary  transmission  line  and  its  load  are  inductive 
the  current  in  the  line  is  a  lagging  one,  and  so  the  power- 
transmitting  capacity  of  the  line  is  less  than  it  should  be.  A 
synchronous  motor  connected  to  such  a  line  may  be  made  to  over- 
come the  inductance  of  the  line  and  its  load  so  that  the  line  has 
its  maximum  capacity.  As  the  excitation  of  the  synchronous 
motor  is  increased,  first  the  inductance  drop  on  the  line  is  over- 
come, and  the  only  drop  is  the  IR  drop.  If  the  excitation  is 
further  increased  the  current  in  the  line  becomes  a  leading  one 
and  will  raise  the  voltage  of  the  generator  by  its  magnetizing 
action.  Also,  if  there  is  a  concentrated  inductance  in  the  line 
near  the  motor,  the  condenser  action  of  the  motor  may  give 
what  is  called  a  "  resonant  rise  "  in  voltage,  resulting  in  a  higher 
E.M.F.  at  the  motor  terminals  than  at  the  beginning  of  the  line. 
This  is  resonance  in  the  same  way  that  an  actual  condenser  and 
inductance  resonate. 

The  "  hunting  "  of  a  synchronous  machine  will  be  briefly 
taken  up  here  because,  in  the  following  tests,  hunting  of  the 
synchronous  motor  may  occur. 

The  hunting  of  a  synchronous  motor  is  generally  the  result  of 
weak  synchronizing  power;  the  phase  position  of  the  motor 
oscillates  about  some  proper  value  of  j8  (for  meaning  of  the  terms 
employed  see  the  discussion  on  parallel  operation  of  alternators, 
Experiment  15),  this  proper  value  of  0  depending  upon  the  load 
the  machine  is  carrying.  Now  if  the  change  in  torque,  for  a 
given  change  of  0,  is  small,  then  the  motor  is  likely  to  oscillate 
violently  around  this  mean  value  of  /3,  and  under  some  conditions 
actually  pulls  itself  out  of  synchronism  with  the  line. 

A  synchronous  motor  is,  in  this  action  of  hunting,  analogous 
to  a  balance-wheel  pendulum.  The  change  in  motor  torque  with 
variation  in  0  (this  change  is  nearly  proportional  to  change  in  /8) 
corresponds  to  the  spring  action  of  the  pendulum,  and  the  mass 
of  the  motor  armature  corresponds  to  the  mass  of  the  balance 
wheel. 

Now  a  balance-wheel  pendulum  has  what  is  called  its  "  natural 


15(3  ALTERNATING  CURRENTS 

period  of  oscillation  "  and  so  has  the  synchronous  motor.  If  a 
periodic  force  of  frequency  equal  to  that  of  the  natural  period  of 
such  a  pendulum  is  impressed  upon  it  the  pendulum  will  oscillate 
with  continually  increasing  amplitude  and  the  system  is  said  to 
be  in  resonance.  The  amplitude  to  which  the  oscillations  will 
build  up  depends  upon  the  value  of  the  dissipative  reactions 
which  are  brought  into  play  by  the  motion  —  if  the  friction  of 
the  system  is  low  very  violent  oscillation  will  result. 

In  the  case  of  the  synchronous  motor  a  similar  occurrence  may 
take  place;  if  the  load,  e.g.,  is  variable  and  changes  in  such  a  way 
as  to  result  in  a  periodic  fluctuation,  the  motor  will  behave  as 
though  it  was  being  acted  upon  by  a  periodic  force.  If  the  period 
of  the  fluctuation  is  near  that  of  the  natural  oscillation  period  of 
the  armature  the  value  of  the  phase  displacement  angle  0  will 
periodically  vary  and  the  armature  oscillates  around  its  normal 
value  of  /3,  which,  of  course,  depends  upon  the  average  load. 
This  oscillation  of  the  armature  is  called  "  hunting."  It  is 
sometimes  so  violent  that  it  is  impossible  to  hold  a  motor  syn- 
chronized with  the  line.  The  amplitude  of  the  hunting  of  the 
synchronous  motor  depends  upon  the  value  of  the  synchronizing 
force  and  upon  the  dissipative  reactions  which  occur  due  to  the 
oscillations.  As  the  value  of  0  periodically  changes,  the  phase 
position  of  the  armature  reaction,  hence  the  position  of  the  field 
flux,  changes  also.  Damping  grids,  so  placed  in  the  pole  faces 
that  this  field  fluctuation  causes  large  eddy-current  losses  in 
them,  tend  to  keep  the  hunting  of  low  amplitude,  and  so  practi- 
cally all  apparatus,  such  as  synchronous  motors  or  converters, 
are  equipped  with  such  preventive  grids. 

If  there  occurs  a  large  drop  in  E.M.F.  in  the  line  feeding  the 
synchronous  machine,  so  that,  with  change  in  current,  there 
occurs  a  large  change  in  e,  the  E.M.F.  impressed  on  the  machine, 
the  synchronizing  effort  of  the  machine  is  correspondingly 
reduced  as  may  be  seen  from  the  equation  given  in  Experiment 


15,  which  was  —  =  —  ^^  —  -  .     This  shows  the  synchronizing 

Op  L  Ka 

force  to  depend  directly  upon  e.     It  also  depends  upon  E,  the 
generated  E.M.F.,  of  the  armature. 

A  synchronous  machine  connected  to  a  high  impedance  line  is, 
therefore,  very  likely  to  hunt,  and  if  the  hunting  is  caused  by  the 
excessive  line  drop  even  damping  grids  will  not  remedy  the 
difficulty.  The  impressed  voltage  of  a  synchronous  machine 


THE  SYNCHRONOUS  MOTOR  157 

must  be  nearly  independent  of  the  current  being  taken  by  the 
machine  if  hunting  is  to  be  eliminated. 

The  hunting  of  a  synchronous  machine  may  sometimes  be 
eliminated  by  changing  its  natural  period;  e.g.,  a  small  rotary 
converter  hunted  so  violently  that  it  would  not  remain  in  syn- 
chronism with  the  line;  a  flywheel  was  mounted  on  the  armature 
shaft,  thus  increasing  the  mass  of  the  oscillating  member  and 
so  changing  its  natural  period;  the  hunting  after  this  addition 
was  imperceptible. 

With  no  load  on  the  synchronous  motor,  and  rated  voltage 
and  frequency  impressed,  reduce  the  excitation  until  150  per  cent 
rated  current  is  flowing  in  the  armature.  Read  impressed  volts 
and  frequency,  amperes  armature  and  field  and  watts  input  to 
armature.  Then  .increase  the  field  excitation  until  about  the 
same  value  of  current  is  obtained  leading  the  impressed  E.M.F., 
and  take  same  readings  as  before.  Get  about  eight  readings  in 
between  these  two  extremes.  Get  a  similar  set  of  readings  for 
half  load  on  the  motor  and  for  full  load.  Take  a  25  per  cent 
overload  run  if  the  motor  will  carry  it.  If  the  motor  has  a 
tendency  to  "  hunt  "  the  readings  for  low  power  factors  will  be 
difficult  to  obtain,  as  under  these  conditions  the  tendency  to 
11  pull  out  "  is  accentuated. 

With  the  synchronous  motor  running  light  and  at  cos  0  =  1, 
impressed  voltage  and  frequency  at  rated  values,  take  a  reading 
of  terminal  volts  and  field  current.  Keeping  the  generator  field 
(i.e.,  the  generator  supplying  power),  constant  and  the  motor 
still  unloaded,  increase  motor-field  current  and  take  another 
reading.  Take  about  eight  similar  readings  with  the  field  of 
motor  superexcited  more  each  time. 

Make  another  set  of  readings  with  a  concentrated  inductance 
in  series  with  the  line  supplying  the  motor.  Read  terminal 
volts,  generator  volts  and  field  current.  In  this  run  the  terminal 
volts  may  rise  above  the  generator  volts  due  to  the  "  resonant  " 
action  of  the  line  and  motor. 

Try  the  effect  on  the  operation  of  the  machine  of  inserting 
some  resistance  in  series  with  the  supply  line. 

Plot  on  one  sheet  the  phase  characteristic  curves,  power  factor, 
and  watts  input  curves  and  on  another  the  curves  of  terminal 
and  generator  voltages.  On  both  sheets  use  field  current  of 
motor  as  abscissae. 


EXPERIMENT   XXIX. 

WITH  MOTOR  EXCITATION  CONSTANT,  TO  FIND  THE  RELATION 
BETWEEN  LOAD  CURRENT  AND  POWER  FACTOR;  PHASE 
SHIFTING  WITH  VARIATION  OF  LOAD. 

IT  is  the  object  of  this  test  to  investigate  the  changes  which 
occur  in  the  armature  current  of  a  synchronous  motor  as  the  load 
is  increased  and  the  motor  C.E.M.F.  is  held  constant;  how  the 
phase  of  the  current  with  respect  to  the  impressed  E.M.F.  alters 
with  load,  and  how  the  phase  position  of  the  armature  changes  as 
the  load  is  varied. 

Referring  to  Fig.  92,  it  will  be  seen  that  if  the  impressed  volt- 
age arid  the  field  current  of  the  motor  are  held  constant,  then 


T 


*.~'-''~~<~l 

90°'  /M        _-" 

^7^     "^^ 

\ 

1***™J                                 T-,                                                        >                     Jf\ 

^j-f-^mCOSOa-^) 

-—  V' 
fr 

\ 

JL 

FIG.  92. 

->-c^    \  / 

>«* 

^'   l\ 

I 
/ 


the  loci  of  EI  and  Em  must  be  circular  arcs  about  0.  The  cur- 
rent, 01,  will  be  used  as  reference  line  and  then  the  resultant 
of  the  motor  E.M.F.  and  line  E.M.F.  must  constantly  be  on 

the  line  OA,  displaced  6  from  01  where  tan  6  =     j;     ,  L  and  R 

t\i 

being  the  constants  of  the  motor  armature.     If  Zm  =  impedance 

OF 
of  motor  armature,  01  =  -~-^' 

4m 

If  the  E.M.F.  relations  are  to  be  investigated  for  a  certain 
load  K,  we  shall  have  the  following  relations : 

K  =  Eml  cos  a     and     Er  =  IZm. 

158 


THE   SYNCHRONOUS  MOTOR 


159 


Now  Er  =  VEm2  -h  E?  +  2  EmEi  cos  («-</>). 

We  have         0  =  (0  -  0)      or      (a  -  0)  +  (0  -  ft)  =  a, 
+  Em  cos  (a  -  0) 


where        cos  /3  = 
Then 
K 


=  Em( 


Er 

Em2+El2+2EmEicos(a-<i>) 


cos  ((a- 


Now  assuming  a  certain  value  for  (a  —  0)  and  knowing  Em,  Et 
and  Zw  it  is  evident  that  we  can  obtain  the  power  output,  input, 
efficiency,  etc.  The  power  factor  of  the  motor  is  at  once  ob- 
tained for  any  value  of  (a  —  $)  by  the  equation 


Although  EI,  Em,  &  and  (a  —  $)  are  known  it  is  not  readily  seen 
how  <£  will  vary,  because  Er  is  itself  an  involved  expression. 


FIG.  93. 

Instead  of  calculating  the  various  quantities  derived  above  they 
may  be  predicted  by  vector  construction.  Assuming  another 
value  of  the  current  =  01',  then  OEr'  =  OF  X  Zm  and  with  Erf 
as  center  and  radius  =  OEm,  cut  the  locus  of  EI  at  EI.  The  <f>' 
is  at  once  obtained  and  Em'I'  cos  a  is  also  determined. 

The  application  of  the  circle  diagram  to  the  synchronous 
motor  is  shown  to  be  possible  in  "  Electric  motors,"  Crocker  & 
Arendt,  from  which  volume  it  is  here  reproduced  in  Fig.  93. 


160  ALTERNATING  CURRENTS 

OEg  is  taken  as  reference  line.  About  0  is  described  a  circle 
with  radius  =  Em  and  another  circle  of  the  same  radius  is 
described  about  the  point  Eg.  This  second  circle  is  the  locus 
of  Er.  The  line  OK  is  drawn  at  the  angle  0  with  OEg  and  OK  = 
OEm'.  The  center  of  the  current-locus  circle  is  obtained  at  B 
by  making  BK  =  OEm.  The  scale  for  the  current  is  obtained 
by  dividing  the  volt  scale  by  Zm.  To  use  this  diagram,  the  angle 
between  Em  and  Eg  is  selected  and  OEm  is  drawn.  EmET  is 
parallel  to  OEg.  With  0  as  center  and  OEr  as  radius  cut  the 
current  locus  at  7,  and  01  represents  the  magnitude  and  phase 
position  of  the  motor  current.  Then  Eml  cos  (f>m  =  motor  out- 
put and  Egl  cos  <£  is  the  value  of  the  input.  Cos  0  is  the  power 
factor  of  the  motor  and  it  may  be  seen  from  the  diagram  that 
the  value  of  <£  for  any  load  depends  upon  the  relative  values  of 
Em  and  Eg  as  was  found  out  in  Experiment  28. 

Adjust  the  motor  voltage  equal  to  that  of  the  line  from  which 
power  is  to  be  taken,  and  synchronize  the  motor.  During  the  test 
keep  the  impressed  voltage  constant.  Read  armature  current, 
watts  input,  watts  output,  phase  position  of  armature  (i.e.,  angle 
|8,  Experiment  15)  and  impressed  voltage.  This  reading  should 
be  taken  with  no  load  on  the  synchronous  motor.  In  case  a 
generator  is  used  for  load  the  connection  diagram  will  be  as 
shown  in  Fig.  94.  The  core  loss  of  the  D.C.  generator  must  be 


Three  phase, 
power  supply 


Load 


FIG.  94. 

measured  at  a  given  generated  voltage  and  its  field  current  kept 
at  the  value  which  gives  this  voltage.  The  resistance  of  the 
generator  armature  must  be  measured  so  that  the  armature  PR 
loss  may  be  calculated. 

By  means  of  the  D.C.  generator  (or  brake)  put  on  J,  J,  f,  full 
and  1J  load  on  the  synchronous  motor  and  take  same  readings 
as  for  no  load.  The  field  of  the  synchronous  motor  is  to  be  kept 


THE  SYNCHRONOUS  MOTOR  161 

constant.  Then  take  a  similar  set  of  readings  for  excitation  of 
the  synchronous  motor  which  give  motor  voltage  25  per  cent 
above  and  25  per  cent  below  normal.  Measure  the  impedance 
and  resistance  of  the  motor  armature. 

Construct  three  circle  diagrams  for  the  three  different  motor 
excitations  and  from  them  obtain  (for  such  values  of  the  angle  <£ 
as  will  give  approximately  the  same  motor  loads  as  those  actually 
measured)  the  values  of  armature  current,  0,  watts  input,  and 
watts  output  of  motor. 

Plot  curves  from  the  experimentally  obtained  results  of  arma- 
ture current,  power  factor  of  motor,  phase  position  of  armature 
and  watts  input  to  motor,  against  watts  output  as  abscissae. 
On  the  same  curve  sheet  plot  the  same  quantities  as  they  are 
obtained  from  the  circle  diagram. 

Instead  of  the  circle  diagram,  the  diagram  given  in  Experiment 
15  serves  excellently  to  analyze  the  action  of  a  synchronous 
motor,  with  varying  excitation  and  load.  The  IZ  drop  in  the 
armature  must  be  reversed  in  phase  to  what  is  shown  in  Experi- 
ment 15,  and  current  phase  will  be  changed,  otherwise  the  con- 
struction is  the  same. 

Note.  —  The  circle  diagram  for  the  synchronous  motor  is  derived  on  the 
supposition  of  constant  impressed  E.M.F.  and  C.E.M.F.  In  the  above  test  the 
impressed  E.M.F.  is  maintained  constant  but  how  nearly  constant  the  C.E.M.F. 
of  the  motor  remains  is  difficult  to  determine.  It  is  evident  that  as  the  phase 
of  the  current  in  the  motor  armature  changes  it  may  have  either  a  magnetizing 
or  demagnetizing  action  upon  the  motor  field.*  In  the  above  test  the  motor- 
field  current  is  maintained  constant  and  the  effect  of  the  armature  reaction  is 
ignored.  This  may,  of  course,  produce  some  discrepancy  between  the  actual 
values  of  the  quantities  plotted  and  then-  values  as  predicted  from  the  circle 
diagram. 

*  For  curves  showing  how  the  variation  of  field  strength  may  change  the 
form  of  the  C.E.M.F.  wave  see  Appendix,  Plate  15. 


EXPERIMENT   XXX. 


STUDY  OF  ROTARY  CONVERTER  RUNNING  FROM  THE  D.C.  END; 

VOLTAGE    RATIOS    FOR    VARIOUS    NUMBER    OF    PHASES; 

VARIATION  OF  VOLTAGE  RATIO  WITH  FIELD  STRENGTH; 

EXTERNAL    CHARACTERISTIC    FOR    INDUCTIVE 

AND  NONINDUCTIVE  LOADS; 

EFFICIENCY. 

THE  voltage  and  current  relations  of  a  rotary  converter  can 
best  be  studied  by  supposing  that  two  different  windings  actu- 
ally exist  on  the  same  armature,  the  two  windings  being  exactly 
similar  as  to  number  of  inductors  and  their  place  upon  the  arma- 
ture core.  One  winding  is  connected  to  slip  rings  and  the  other 
to  a  commutator.  First  a  single-phase  converter  (one  having 
taps  180°  apart  and  which  really  is  a  two-phase  winding)  will  be 
considered.  •  Suppose  there  are  eight  coils  in  each  winding  and 
the  machine  is  bipolar.  When  the  arma- 
ture is  rotating  each  coil  will  generate  a 
sine  wave  of  E.M.F.  (if  the  field  distribu- 
tion is  suitable)  and  these  E.M.F. 's  will 
differ  in  time  phase  by  the  same  amount  as 
their  respective  coils  differ  in  space  phase.* 
The  vector  representation  of  these  E.M.F. 's 


FIG.  95. 

is  given  in  Fig.  95  (a).  'As  the  coils  are  connected  in  series  the 
E.M.F.  in  the  whole  armature  circuit  is  zero  as  shown  in  Fig. 

*  If  the  number  of  coils  is  sufficient  (say  three  or  more  per  phase  per  pair  of 
poles)  the  E.M.F.  relations  obtained  in  the  following  discussion  hold  good  even 

162 


THE  ROTARY  CONVERTER  163 

95  (6),  If,  however,  the  winding  is  tapped  at  two  connections 
180°  apart  and  these  taps  connected  to  slip  rings  it  is  evident 
that  there  will  be  an  alternating  E.M.F.  on  the  rings,  the  max- 
imum value  of  which  is  the  diagonal  of  the  octagon.  This  max- 
imum value  occurs  when  the  points  tapped  to  the  rings  are  hi 

the  neutral  plane  of  the  magnetic  field.     The  effective  value 

pi 

of  this  alternating  E.M.F.  will  be  —~  where  Em  =  diagonal  of 

octagon. 

Now  the  second  winding  is  connected  to  a  commutator  and 
the  brushes  continually  connect  to  the  taps  which  are  temporarily 
in  the  neutral  plane,  so  that  the  E.M.F.  at  the  brushes  is  Em. 
If  now  the  armature,  instead  of  being  driven  by  some  outside 
power,  is  used  as  a  synchronous  motor,  power  being  supplied  to 
the  collector  rings,  the  ratio  of  the  voltage  on  the  D.C.  winding 
to  the  impressed  E.M.F.  will  be  practically  V2 : 1,  for  when  the 
impedance  drop  in  the  A.C.  winding  is  small  (which  it  generally 
is)  the  impressed  and  counter  E.M.F.'s  are  nearly  equal. 

The  current  which  flows  in  the  A.C.  winding  will  be  nearly 
in  phase  with  the  impressed  E.M.F.,  which  means  that  it  is  180° 
out  of  phase  with  the  vector  representing  the  E.M.F.  generated 
by  the  group  of  coils  connected  in  series  between  taps.  The 
current  in  each  coil  will  be  a  sine  wave. 

If  now  a  current  is  taken  from  the  D.C.  winding  the  current 
will  flow  in  phase  with  the  voltage  at  the  brushes  (i.e.,  in  phase 
with  the  generated  voltage  of  the  group  of  coils  on  one  side  of  the 
armature)  and  this  means  that  it  is  in  phase  with  the  C. E.M.F. 
of  the  synchronous  motor  and  hence  180°  out  of  phase  with  the 
current  in  the  A.C.  winding.  Also  the  current  in  the  D.C. 
winding  is  not  a  sine  function  but  has  a  constant  value  for  one- 
half  a  revolution  of  the  armature,  reversing  its  direction  to  a 
negative  quantity  (of  same  magnitude  as  before)  just  as  the  coil 
undergoes  commutation. 

Now,  if  the  two  windings  are  one,  the  same  winding  being  con- 
though  the  wave  form  of  the  E.M.F.  generated  in  a  single  coil  is  widely  differ- 
ent from  a  sine  wave.  An  irregular  wave,  such  as  is  generally  developed  in  a 
single  coil,  signifies  the  existence  of  upper  harmonics  in  the  E.M.F.  wave  and 
these  higher  harmonics  tend  ordinarily  to  neutralize  one  another  when  several 
differently  placed  coils  are  connected  in  series  with  each  other,  as  they  are  on 
a  rotary  armature.  So  that  with  the  E.M.F.  generated  per  coil  an  irregular 
curve  the  form  of  the  voltage  wave  between  180°  or  120°  taps  of  the  rotary 
approximates  closely  to  a  sine  wave. 


164 


ALTERNATING  CURRENTS 


nected  to  slip  rings  on  one  end  and  to  a  commutator  on  the  other, 
exactly  the  same  relations  of  E.M.F.  and  current  will  be  true. 
The  ratio  of  A.C.  to  D.C.  voltage  is  as  1:  V2,  and  in  each  coil 
there  will  be  the  resultant  of  a  sine  curve  alternating  current  and 
a  constant  current  which  reverses  its  direction  once  for  each 
alternation  of  the  A.C.  current.  As  this  D.C.  reverses  in  the 
different  coils  at  different  times  while  the  A.C.  current  has  the 
same  phase  in  all  coils,  it  is  evident  that  this  resultant  current 
will  be  different  for  the  different  coils.  As  the  rating  of  a  ma- 
chine depends  altogether  upon  its  heating,  and  the  heating  upon 
the  current,  it  is  important  to  consider  here  more  in  detail  this 
resultant  current.  An  eight-coil  armature  will  be  considered 

and  the  different  coils  num- 
bered for  identification  as 
in  Fig.  96.  The  effect  of 
losses  in  the  machine  itself 
will  be  neglected,  so  that 
output  =  input.  The  slip- 
rings  are  represented  out- 
side the  commutator  and 
time  is  reckoned  zero  when 
coil  1  is  just  entering  the 
neutral  plane,  i.e.,  the  D.C. 
component  of  current  in 
No.  1  is  just  going  to  re- 
verse. As  the  A.C.  taps  are  also  in  the  neutral  plane,  the  A.C. 
C.E.M.F.,  and  hence  the  A.C.  current,  are  at  their  maximum 
values. 

Now  if  the  D.C.  voltage  =  100,  the  A.C.  voltage  (effective)  = 
70.7;  and  as  output  =  input,  if  we  assume  a  direct  current  of  10 
amperes,  the  maximum  A.C.  current  will  be  20  amperes.  These 
line  currents  are  twice  as  large  as  the  coil  currents,  the  armature 
being  two  circuit.  Then  just  before  commutation  the  current  in 
coil  1  =  10  —  5  =  5.0  amperes  and  just  after  commutation  (the 
A.C.  current  still  being  a  maximum)  the  current  in  the  coil  = 
10  -|-  5  =  15.0  amperes.  After  the  armature  has  revolved  45° 
the  A.C.  current  has  a  value  of  10  X  cos  45°  =  7.07  amperes, 
so  at  this  instant  current  in  coil  No.  1  (and  also  2,  3  and  4)  = 
7.07  +  5  =  12.07  amperes;  but  coils  8,  7,  6  and  5  will  have 
7.07  —  5  =  2.07  amperes  at  this  time.  At  time  =  90°  the 
A.C.  current  =  0,  so  that  all  coils  have  5  amperes.  By  taking 


FIG.  96. 


THE  ROTARY  CONVERTER 


165 


successive  intervals  of  time  in  this  fashion  and  finding  the  current 
in  each  coil  it  will  be  found  that  the  current  distribution  is  not 
regular,  those  near  the  A.C.  taps  carrying  much  more  current 
than  .the  others;  the  form  of  current  wave  for  two  typical  coils  is 
shown  hi  Fig.  97.*  For  this  reason  the  heating  of  the  armature 


Coil  midway  between 


A.C.  taps 


Coil  at  A.C.  tap 


FIG.  97. 

of  a  rotary  converter  is  irregular,  those  coils  next  to  the  A.C. 
taps  becoming  the  hottest.  If,  however,  the  A.C.  current  is  not 
in  phase  with  the  impressed  E.M.F. 
then  the  hottest  coils  will  not  be 
those  adjacent  to  the  A.C.  taps  but 
will  be  more  towards  the  center  of  the 
group.  Also,  for  a  given  load,  the  ac- 
tual value  of  the  current  in  all  the 
coils,  and  hence  the  heating,  will  be 
greater  than  where  cos  <£  =  1. 

If  now  we  consider  a  polyphase 
converter  instead  of  a  single-phase, 
the  ratio  of  A.C.  to  B.C.  voltage  will 
evidently  be  different.  For  a  bipolar 
machine  the  D.C.  voltage  is  equal  to 

the  vector  sum  of  the  E.M.F.'s  generated  by  one-half  of  the  coils 
as  before.  But  the  A.C.  voltage  will  depend  upon  how  many  coils 
are  included  between  taps.  If,  e?g.,  a  three-phase  converter  is 

*  For  experimental  determination  of  these  current  forms  see  Appendix, 
Plates  20-23. 


166  ALTERNATING  CURRENTS 

considered  it  is  evident  that  this  A.C.  voltage  will  be  different 
than  that  of  a  single-phase  converter  of  the  same  D.C.  voltage. 
For  the  three-phase  machine  the  A.C.  voltage  may  be  found  as 
indicated  in  Fig.  98.  Construct  a  circle  having  for  diameter 

=  -^=D.C.   volts.     This  diameter  represents  the  A.C.  voltage 

for  a  single-phase  rotary  where  the  taps  are  180°  apart.  When 
the  taps  are  120°  apart  the  effective  A.C.  voltage  between  taps  is 
equal  to  the  chord  AD,  because  the  circumference  represents  the 
circumscribed  circle  of  the  polygon  formed  by  the  vector 
E.M.F.'s  of  all  the  armature  coils,  and  one-third  of  the  armature 
coils  add  (vectorially)  their  E.M.F.'s  to  give  the  voltage  AD. 
If  a  six-phase  converter  is  used  the  voltage  between  adjacent 
taps  is  equal  to  the  chord  AE. 

If  a  —  angle  between  taps  it  is  clear  that 

A.C.  voltage  =  [— =D.C.  voltage  )  sin ^ • 
\v2  /       2 

As  the  number  of  taps  (i.e.,  the  number  of  phases)  increases, 
the  heating  of  the  armature  will  decrease  due  to  the  fact  that  the 
mean  resultant  current  in  the  different  coils  decreases  with 
increase  of  taps.  A  given  rotary  having  a  capacity  of  1  used 
three-phase  will  have  a  capacity  of  1.44  where  used  six-phase 
and  a  capacity  of  0.75  if  used  as  a  D.C.  generator.  (For  a  more 
complete  discussion  of  the  heating  of  rotary  converters  the 
student  is  referred  to  "  A.C.  Motors  "  by  McAllister.) 

As  the  A.C.  voltage  bears  a  constant  relation  to  the  D.C. 
voltage  it  is  seen  that  with  constant  impressed  D.C.  voltage  the 
A.C.  voltage  cannot  be  varied  by  changing  the  field  strength. 
If,  e.g.,  the  field  is  decreased  to  J  normal  value  the  speed  of  the 
rotary  (running  inverted)  will  increase  to  twice  normal  value 
and  so  the  A.C.  voltage  will  remain  constant.  If,  however,  the 
D.C.  voltage  is  changed,  the  A.C.  will  change  in  the  same  ratio. 
It  has  been  said  that  the  ratio  of  A.C.  to  D.C.  volts  is  constant, 
but  this  is  only  true  when  the  armature-impedance  drop  is 
negligible.  As  the  load  on  an  inverted  rotary  increases  its 
C.E.M.F.  decreases  and  so  the  terminal  E.M.F.  on  the  A.C.  side 
will  decrease. 

When  the  alternating  current  taken  from  the  rotary  is  in  phase 
with  the  generated  voltage  there  will  be  no  appreciable  change 
in  field  strength  due  to  the  armature  reaction,  because  the  D.C. 


THE  ROTARY  CONVERTER  167 

and  A.C.  reactions  just  neutralize  one  another.  If,  however,  the 
A.C.  load  is  inductive,  armature  reaction  will  demagnetize  the 
field  and  hence  the  speed  must  increase  to  keep  the  C.E.M.F. 
on  the  D.C.  end  nearly  equal  to  the  impressed  E.M.F.  In  some 
cases  this  field  weakening  may  be  great  enough  to  cause  the 
rotary  to  run  away.  This  is  likely  to  occur  when  an  induction 
motor  is  started  with  power  furnished  by  an  inverted  rotary  of 
somewhere  near  the  same  size  as  the  motor. 

For  a  given  size  machine  the  stray-power  and  field  losses  will 
be  approximately  the  same  whether  it  is  used  as  a  D.C.  machine, 
an  A.C.  machine,  or  a  rotary  converter.  The  armature  PR  losses 
will  be  considerably  less  in  the  last  case,  however,  so  that  hi  general 
a  rotary  converter  is  slightly  more  efficient  than  either  an  A.C.  or 
D.C.  generator  of  the  same  size. 

For  a  given  impressed  D.C.  voltage,  find  the  A.C.  voltage 
between  each  and  every  tap  of  a  single-,  three-,  quarter-  and  six- 
phase  rotary  converter.  Construct  a  circle  having  a  diameter  = 

D.C.  volts  ,. 

— -= and  check  the  vector  diagram  given  in  the  previous 

discussion. 

With  a  fixed  D.C.  voltage  vary  the  field  strength  through  as 
wide  a  range  as  is  permissible  and  read  D.C.  volts,  A.C.  volts 
and  field  current  and  speed.  Explain  results  of  these  two  tests 
and  plot  curves  to  show  relations  between  variables  involved. 

With  constant-rated  impressed  D.C.  volts,  and  field-current 
constant  at  normal  value,  load  the  converter  on  the  A.C.  side  with 
noninductive  load.  Read  D.C.  volts,  A.C.  volts,  field-current, 
D.C.-current  and  A.C.-current  speed.  Take  readings  at  about 
six  different  loads  between  zero  and  50  per  cent  overload.  Take 
a  similar  run  with  an  inductive  load,  using  wattmeter  on  A.C. 
side  to  get  power  factor.  Maintain  power  factor  constant  at 
about  0.8.  Care  must  be  observed  that  the  safe  speed  is  not 
exceeded. 

From  data  of  these  two  runs  calculate  efficiency  of  rotary. 

Plot  curves  of  A.C.  volts,  speed  and  efficiency,  against  A.C. 
current  output  as  abscissa. 


EXPERIMENT  XXXI. 

ROTARY  CONVERTER  RUNNING  FROM  A.C.  END;  STARTING  BY 

VARIOUS  METHODS;  EXTERNAL  CHARACTERISTIC,  WITH 

AND  WITHOUT  SERIES  FIELD  ON  INDUCTIVE  LINE 

AND   NONINDUCTIVE  LINE. 

THERE  are  various  methods  employed  for  starting  rotary  con- 
verters and  getting  them  to  the  proper  speed  for  synchronizing 
with  the  A.C.  power  supply. 

If  a  suitable  source  of  D.C.  power  is  available  the  rotary  may 
be  started  from  the  D.C.  end  as  a  shunt  motor.  Where  the 
rotaries  already  running  are  supplying  a  lighting  system  the 
voltage  on  the  D.C.  line  will  be  fairly  constant  and  the  incoming 
rotary  may  be  started  from  the  D.C.  bus  bar.  If  the  D.C.  load 
is  railway  work  the  sudden  and  wide  fluctuations  in  power  con- 
sumption cause  the  bus  voltage  to  vary  and  hence  the  speed  of 
the  incoming  rotary  being  started  from  such  a  line  will  vary  and 
the  process  of  synchronizing  is  more  difficult.  Where  a  station 
is  equipped  with  a  storage  battery  to  carry  peak  loads  this 
battery  is  a  convenient  source  of  power  for  starting  the  rotaries. 
If  the  substation  is  completely  shut  down  the  first  rotary  must 
be  started  from  storage  battery  or  else  one  of  the  methods  for 
starting  from  the  A.C.  end  may  be  employed.  If  several  sub- 
stations are  operating  in  parallel  on  a  distribution  system  the 
first  rotary  in  any  station  may  be  started  from  the  D.C.  end  with 
power  furnished  by  one  of  the  other  stations. 

Probably  the  best  way  to  start  up  a  rotary  is  to  have  a  small 
polyphase  induction  motor  directly  connected  upon  the  shaft  of 
the  converter.  This  motor  need  only  be  large  enough  to  fur- 
nish the  normal  stray-power  losses  of  the  rotary  and  must  have 
such  a  speed-load  characteristic  that  when  furnishing  power 
enough  to  supply  the  no-load  losses  of  the  rotary,  it  gives  the 
proper  speed  to  make  the  rotary  run  in  synchronism  with  the  line. 
This  will  always  necessitate  the  induction  motor  having  at  least 
one  pair  of  poles  less  than  the  rotary.  It  is  impossible  to  design 
an  induction  motor  so  accurately  that  its  speed,  when  carrying 
the  load  necessary  to  run  with  the  rotary  light,  is  exactly  the 

168 


THE  ROTARY  CONVERTER  169 

speed  demanded  by  the  rotary.  If  its  speed  is  slightly  too  high 
it  may  be  reduced  by  increasing  the  resistance  of  its  rotor  end 
rings  or  by  putting  a  small  resistance  in  two  of  the  leads  (on  a 
three-phase  motor)  supplying  its  power. 

For  the  process  of  synchronizing  small  changes  of  speed  are 
necessary  to  get  correct  phase,  etc.  These  slight  changes  may 
be  accomplished  by  altering  the  field  strength  of  the  rotary,  thus 
changing  its  core  losses.  This  change  will  result  in  the  rotary  not 
having  exactly  the  same  A.C.  voltage  as  the  line  with  which  it 
is  to  be  synchronized,  but  generally  this  will  do  no  harm.  Its 
only  effect  is  that  the  current  taken  by  the  rotary  upon  synchro- 
nizing is  somewhat  larger  than  it  should  be.  After  the  rotary 
has  been  switched  on  to  the  A.C.  line  the  power  circuit  to  the 
induction  motor  is  opened  and  its  rotor  allowed  to  run  free.  As 
there  will  be  no  flux  in  the  motor  the  continuous  motion  of  its 
rotor,  while  the  converter  is  operating,  requires  practically  no 
power. 

In  using  either  of  the  methods  so  far  described  a  synchronoscope 
or  lamps  may  be  used  to  determine  the  proper  instant  for  closing 
the  synchronizing  switch.  The  lamps  may  be  connected  for 
either  "  dark  "  or  "  light  "  synchronizing,  being  cross  connected 
for  the  latter  and  straight  across  switch  blades  for  the  former. 
If  the  lamps  used  on  a  polyphase  converter  do  not  flicker  in  phase 
with  one  another,  one  of  the  phases  on  the  rotary  is  reversed  and 
a  pair  of  the  supply  lines  must  be  reversed.  Referring  to  Fig.  99, 
if  the  three  lamps  a,  b,  c  flicker  simultaneously  the  supply  line 


Three  phase 
power 


is  properly  connected  to  the  rotary,  but  if  they  brighten  in 
rotation,  as  a — b — c,  then  two  of  the  leads  A,  B,  C,  must  be 
interchanged. 

As  has  been  remarked  in  one  of  the  experiments  upon  the  syn- 
chronous motor,  a  polyphase  synchronous  machine  may  be 
started  as  an  induction  machine  if  its  field  is  left  without  excita- 
tion and  a  reduced  voltage  is  applied  to  its  armature.  The 
current  taken  is  excessive  but  not  enough  so  to  overheat  the 


170  ALTERNATING  CURRENTS 

machine  during  the  short 'time  necessary  for  starting.  In  start- 
ing by  this  method  a  field  break-up  switch  must  be  used,  other- 
wise the  field  coils,  acting  as  the  secondary  winding  of  a  step-up 
transformer  (the  armature  coils  being  the  primary),  may  generate 
sufficient  voltage  to  break  down  their  insulation. 

The  recent  tendency  in  rotary  converter  design  has  been  towards 
higher  voltage  machines,  1200  volt  rotaries  now  being  used.  This 
high  voltage  necessitates  the  use  of  commutating  poles  on  the 
rotary,  as  the  voltage  per  coil  is  higher  than  with  the  600  volt 
machine,  and  therefore  commutation  is  more  difficult  to  accom- 
plish. Now,  the  use  of  commutating  poles  brings  about  a  difficulty 
when  the  rotary  is  started  as  an  induction  motor.  As  the  poly- 
phase field,  developed  by  the  currents  of  the  rotary  armature, 
passes  the  commutating  poles,  owing  to  the  low  reluctance  of  the 
path  of  the  leakage  flux  (the  face  of  the  commutating  pole  forming 
part  of  the  magnetic  path)  a  very  heavy  current  will  flow  in  that 
coil  which  is  short-circuited  by  the  brushes  in  the  D.C.  end  of  the 
machine  and  violent  sparking  at  the  brush  contacts  will  result. 
This  sparking  is  so  pronounced  in  the  starting  of  commutating- 
pole  rotaries  by  the  induction  motor  principle,  that  the  method 
cannot  be  employed  unless  some  special  precautions  are  observed. 
One  type  of  high-voltage  interpole  rotary  has  been  designed  with 
a  brush-lifting  mechanism;  while  the  machine  is  being  started  all 
of  the  D.C.  brushes  are  lifted  from  the  commutator,  then  after  the 
machine  has  reached  synchronous  speed  they  are  lowered  into  their 
proper  place  on  the  commutator  by  a  simple  lever  action. 

The  induction-motor  method  of  starting  will  fail  if  the  poles  are 
laminated  and  no  damping  grids  are  used  on  the  pole  faces. 
However,  as  heavy  grids  are  necessary  to  prevent  hunting  of  the 
rotary  this  method  of  starting  may  nearly  always  be  used,  but  the 
large  starting  current  necessary  disturbs  the  line  voltage  to  a 
considerable  extent  and  may  even  throw  out  of  step  other  ma- 
chines already  synchronized. 

When  this  method  of  starting  is  employed  the  transformers  are 
provided  with  low-voltage  taps  and  a  double-throw  switch  is 
used.  The  low  voltage  is  applied  to  the  armature  until  it  reaches 
nearly  synchronous  speed,  when  the  switch  is  thrown  over  and 
the  armature  is  connected  to  the  power  supply  of  normal  voltage, 
when  it  will  generally  pull  into  synchronous  speed.  The  polarity 
of  the  D.C.  end  of  the  rotary,  as  shown  by  the  voltmeter,  may, 
however  be  reversed ;  this  signifies  that  the  rotary  has  slipped  into 


THE  ROTARY  CONVERTER  171 

synchronous  speed  180°  out  of  its  proper  phase  and  some 
method  (as,  e.g.,  restarting)  must  be  used  to  make  it  slip  back 
one  pole.  If  the  rotary  has  come  into  synchronism,  in  the  correct 
phase,  the  field  current  may  be  gradually  increased  until  such  a 
value  is  reached  as  makes  the  armature  current  a  minimum. 

Ordinarily  the  voltage  on  the  D.C.  end  of  a  rotary  will  fall 
as  the  load  is  increased,  for  two  reasons.  The  impedance  drop  of 
the  line  increases  with  increase  of  load  as  does  also  the  imped- 
ance drop  in  the  armature  of  the  rotary.  Hence,  if  a  shunt-wound 
rotary  is  connected  to  an  A.C.  power  line  the  voltage  of  which  is 
maintained  constant  at  the  generator  end,  the  D.C.  voltage  of  the 
rotary  will  fall  off  considerably  as  load  is  applied,  the  drop  being 
proportional  to  the  armature  impedance  and  line  impedance. 

As  will  be  readily  appreciated,  if  the  line  supplying  power  to 
the  rotary  has  an  extra  inductance  inserted  the  drop  in  D.C. 
volts  will  be  greater  than  before,  for  same  values  of  loads. 

So  far  we  have  considered  a  rotary  running  with  constant  field 
strength,  i.e.,  constant  C.E.M.F.  Now  a  rotary  field  may  employ 
series-field  excitation  as  well  as  shunt-field  excitation,  in  which 
case  the  field  strength,  hence  the  C.E.M.F.  of  the  rotary,  will 
increase  with  load.  In  the  discussion  of  the  synchronous  motor 
it  was  shown  that  if  the  C.E.M.F.  of  a  synchronous  machine  is 
greater  than  the  E.M.F.  of  the  line  supplying  its  power,  then  the 
current  taken  by  such  a  machine  will  lead  the  impressed  E.M.F. 
by  a  certain  angle,  the  value  of  which  depends  upon  the  amount 
of  superexcitation  of  the  field  of  the  synchronous  machine. 

A  rotary  converter  having  series-field  excitation,  being  supplied 
with  power  through  an  inductive  line,  tends  to  automatically 
compound  itself  as  the  load  increases;  but  it  has  been  shown  in 
Experiment  30  that  the  ratio  of  D.C.  volts  to  A.C.  volts  in  a 
converter  is  constant  provided  that  there  is  no  field  distortion 
(this  question  of  field  distribution  will  be  investigated  in  Experi- 
ment 32).  Hence  it  must  be  that  if  the  D.C.  volts  of  the  rotary 
increase  with  the  load  the  A.C.  voltage  impressed  must  cor- 
respondingly  increase.  This  is  what  actually  occurs,  and  the 
cause  of  such  rise  in  A.C.  voltage  at  the  end  of  an  inductive 
transmission  line  will  be  seen  from  the  accompanying  vector 
diagrams. 

Suppose  that  the  rotary  has  such  field  excitation  that  the  cur- 
rent fed  into  the  transmission  line  is  in  phase  with  the  generator 
voltage.  Then  the  E.M.F.  relations  will  be  as  represented  in 


172 


ALTERNATING  CURRENTS 


FIG.  100. 


Fig.  100,  in  which  OG  represents  the  magnitude  of  the  impressed 

voltage  and  the  phase  of  the  cur- 
rent. The  impedance  drop  of  the 
line  is  OZ  and  the  A.C.  voltage 
impressed  on  the  rotary  is  obtained 
by  subtracting  this  from  OG,  giving 
the  rotary  voltage  OC,  which  is 
somewhat  less  than  OG.  If  now  the 
rotary  is  much  underexcited  so  that 
the  current  in  the  transmission  line 

lags  behind  the  generator  volts  the  rotary  volts  will  be  given  by 

the  construction  in  Fig.  101,  in  which  the  letters  have  the  same 

significance  as  in  Fig.  100.     Here  it 

is  seen  that  the  rotary  voltage  is 

much  less  than  if  the  current  is  made 

to  lead  the  generator  volts  by  over- 
exciting  the  rotary;  in  such  case 

the  rotary  volts  may  be  larger  than 

the  generator  volts,  as  shown  in  Fig.  102. 


o 


FIG.  101. 


Now,  if  the  rotary 


will  maintain  at  all  loads  a 
leading  current  it  is  evident 
that  the  machine  will  be  com- 
pounding. In  Fig.  103,  for 
example,  at  no  load  the  A.C. 
voltage  on  the  rotary  is  rep- 
resented by  OA .  At  half  load 
it  increases  to  OB  and  at  full 
load  is  OC.  If  the  angle  <£ 
should  decrease  with  increase  of  load  then  the  compounding  will 
not  be  so  marked,  and  if  <j>  increases  with  load,  compounding  will 


FIG.  102. 


FIG.  103. 

exist  to  a  greater  extent  than  if  0  remains  constant  from  no  load 
to  full  load.     If  there  are  enough  turns  in  the  series  field  the 


THE  ROTARY  CONVERTER  173 

angle  <£  may  be  lagging  at  no  load  and  when  the  load  increases 
it  will  gradually  change  to  a  leading  angle.  Under  such  con- 
ditions, compounding  will  exist  to  the  greatest  degree  possible. 

As  may  be  seen  from  this  discussion  and  the  vector  diagrams, 
for  a  rotary  to  be  automatically  compounding  (not  considering 
the  synchronous  booster)  it  must  be  operating  on  an  inductive 
line  and  have  sufficient  series  turns  to  overcome  the  inductance 
of  the  line.  The  amount  of  compounding  depends  upon  the 
amount  of  inductance  in  the  line  and  the  strength  of  the  series 
field. 

It  should  be  noted  that  the  use  of  an  overexcited  rotary  for 
neutralizing  lagging  current,  existing  in  the  transmission  line  to 
which  the  rotary  is  connected,  is  not  generally  advisable.  As  has 
been  pointed  out  the  heating  in  the  coils  of  a  rotary  is  very  un- 
equal, and  although  the  machine,  as  a  whole,  may  be  running  cool 
enough  some  few  coils  may  be  overheated.  This  possibility  is 
much  exaggerated  if  the  rotary  is  made  to  operate  at  power  fac- 
tors other  than  one. 

In  the  laboratory  to  get  an  inductive  line  the  rotary  is  supplied 
through  a  concentrated  inductance.  In  this  case  a  still  greater 
rise  of  E.M.F.  at  the  rotary  may  occur,  due  to  a  kind  of  resonance 
between  the  inductance  and  the  superexcited  rotary.  In  this 
case  the  terminal  voltage  of  the  rotary  may  become  larger  than 
the  E.M.F.  of  the  generator  supplying  power  to  the  rotary. 

Try  the  various  methods  of  starting  a  rotary;  when  using 
lamps  to  synchronize,  try  both  connections.  Explain  the  effect 
observed  when  the  phases  are  incorrectly  connected  for  synchro- 
nizing. With  constant  generator  voltage,  obtain  the  external 
characteristic  of  the  rotary  on  inductive  and  on  a  noninduc- 
tive  line,  without  the  use  of  the  series  field.  Obtain  similar 
curves  when  the  series  field  is  used  to  compound  the  rotary. 
Keep  the  shunt-field  circuit  resistance  constant  throughout  the 
test,  at  normal  value.  For  all  curves  read  input  (amperes,  volts 
and  watts)  and  output,  also  generator  voltage. 

Plot  curves  of  D.C.  volts,  impressed  A.C.  volts,  power  factor 
and  efficiency,  using  D.C.  amperes  load  as  abscissa.  If  the  same 
rotary  is  used  in  this  test  as  hi  Experiment  30,  explain  any 
discrepancy  in  the  efficiency  curves  of  the  two  tests. 


EXPERIMENT   XXXII. 

STUDY  OF  THE  AUXILIARY  POLE  ROTARY  CONVERTER;  VARIA- 
TION OF  VOLTAGE  RATIO  WITH  DIFFERENT  FIELD  EX- 
CITATIONS AND  EXAMINATION  OF  FIELD  FORM 
TO  CHECK  VOLTAGE  RATIOS. 

IN  the  auxiliary-pole  rotary  the  compounding  is  obtained  by 
altering  the  field  from  its  normal  distribution.  The  method 
can  be  applied  only  to  three-  or  six-phase  converters,  the  reason 
for  which  will  become  apparent  later. 

The  D.C.  voltage  of  any  generator  depends  upon  the  average 
value  of  the  field  under  an  entire  pole  face.  The  exact  distribution 
of  the  flux  is  of  no  moment.  Whether  the  flux  density  is 
uniform  or  not  under  a  pole  face  will  produce  no  difference  in  the 
D.C.  voltage  generated  provided  the  average  flux  density  is  the 
same  in  both  cases. 

In  the  same  way,  when  an  A.C.  winding  is  considered  with 
taps  120°  apart,  the  distribution  of  the  flux  in  the  120°  spanned 
by  the  coil  is  not  of  so  much  importance  in  determining  the  value 
of  the  E.M.F.  generated.  The  field  distribution  will  alter  the 
wave  form  somewhat,*  but  in  a  three-phase  winding  this  de- 
parture from  a  sine  wave  is  exceedingly  small  even  when  the 
field  has  an  excessive  distortion.  The  factor  which  must  be  ob- 
served to  maintain  constant  generated  alternating  E.M.F. is  that 
the  maximum  flux  which  can  be  embraced  by  a  coil  remains  con- 
stant, i.e.,  if  the  flux  density  is  increased  in  one  part  of  the  space 
covered  by  a  coil,  it  must  be  correspondingly  decreased  in  an- 
other. Suppose  then  that  the  pole  of  the  converter  is  made  in 
three  parts  (in  which  form  it  was  first  projected),  the  excitation 
of  each  part  being  under  separate  control.  Then  if  all  three 
parts  are  magnetized  equally  the  field  distribution  may  be 
represented  as  by  the  full-line  diagram  in  Fig.  104.  The  average 
value  of  this  field  is  X  and  hence  X  may  be  taken  as  a  measure 
of  the  D.C.  voltage  generated  and  also  of  the  A.C.  voltage. 
Now  if  the  part  B  is  weakened  and  the  part  A  and  C  both 

*  This  change  is  very  slight  as  shown  by  the  results  given  by  Plate  27  of 
the  Appendix. 

174 


THE  ROTARY  CONVERTER  175 

strengthened  by  an  equal  amount  it  will  be  evident  that  the 
total  flux  under  the  pole  face  has  been  increased  and  hence 
the  average  flux  density  has  been  increased.  The  value  of  the 
average  density  is  now  given  by  the  line  Y  and  so  Y  is  a  meas- 
ure of  the  D.C.  voltage 
which  would  be  gener- 
ated by  such  a  field. 
But  it  will  be  noticed 
that  the  average  density 
for  that  part  of  the  field 

embraced  in  120°  of  arc     YX    jj       Magnetic  Flux  Distribution 

(i.e.,    distance    between 


n 


A.C.   taps)    is   just   the      *  m         isoo    Jj 

same   as   it   was  before      « 

^  ij  j  F10-  104- 

the  field  was  made  non- 
uniform.  This  is  due  to  the  fact  that  120°  includes  only  two  of 
the  pole  parts  and  one  part  is  strengthened  just  as  much  as  the 
other  is  weakened.  Hence  the  C.E.M.F.  generated  between  A.C. 
taps  by  the  machine  will  be  the  same  with  this  irregular  distri- 
bution as  it  was  for  uniform  field,  and  as  the  C.E.M.F.  is  practi- 
cally of  the  same  magnitude  as  the  impressed  voltage  it  is  seen 
that  a  three-phase  rotary,  designed  for  a  certain  normal  field, 
will  operate  just  as  well  with  a  distorted  field,  provided  that  the 
distortion  introduced  does  not  change  the  total  flux  embraced 
by  the  windings  of  one  phase  and  that  the  distortion  is  not 
sufficient  to  change  the  form  of  the  C.E.M.F.  wave.  If  such  a 
distortion  was  introduced  that  the  wave  form  of  the  E.M.F. 
generated  was  changed,  then  magnetizing  currents  would  flow 
from  the  line  into  the  armature  and  would  immediately  neu- 
tralize the  distorting  M.M.F.  and  bring  the  field  back  to  its 
proper  form. 

The  distorted  field  represented  by  the  dotted  line  in  Fig.  104, 
however,  does  change  the  D.C.  volts  generated  and  changes  this 
voltage  in  the  ratio  of  Y  to  ^T.  Therefore,  it  becomes  evident 
that  by  suitably  distorting  the  field  of  a  rotary  the  A.C.  end  will 
operate  as  though  no  distortion  occurred,  but  the  D.C.  voltage 
will  be  raised. 

If  the  middle  section  of  the  pole  was  strengthened  and  the 
two  outside  sections  weakened  by  an  equal  amount  the  A.C. 
voltage  would  still  be  the  same  but  the  D.C.  voltage  would  be 
smaller  than  X. 


176 


ALTERNATING  CURRENTS 


If  the  A.C.  taps  were  180°  apart  (a  two-  or  four-phase  con- 
verter) this  field-distortion  compounding  method  could  not  be 
used  because  both  the  magnitude  and  form  of  the  A.C.  wave 
generated  in  180°  of  the  winding  will  change  when  the  field  is 
distorted  as  above  described.* 

To  make  each  field  pole  in  three  parts  produces  a  cumbersome 
and  expensive  machine.  It  has  been  found  that  the  idea  of  a 
compounding  by  field  distortion  can  be  carried  out  with  two-part 
poles  practically  as  well  as  though  three  parts  were  used.  One 
section  of  the  pole  is  considerably  larger  than  the  other.  As  a 
two-part  pole  can  be  made  much  smaller  and  less  expensive  than 
a  three-part,  all  split-pole  rotaries  are  at  present  made  with  the 
two-part  poles.  In  the  two-part  pole  construction,  however, 
more  difficulty  is  experienced  with  the  shifting  of  the  commuta- 
ting  plane  as  the  field  undergoes  distortion.  The  three-part  pole 
gives  practically  no  shift  as  the  field  is  distorted.  By  the  use  of 
commutating  poles  this  difficulty  of  commutation  is  readily 
overcome  in  the  two-part  pole  construction. 

Owing  to  the  broken-up  character  of  the  magnetic  path  of  this 
type  of  converter  it  will  not  synchronize  readily  when  started 
from  the  A.C.  end  as  an  induction  motor.  Such  a  converter  is 
very  likely  to  stick  at  one-third  or  one-half  synchronous  speed 
and  not  accelerate  any  more. 

The  two-part  pole  and  its  possible  field  forms  are  shown  in 
Fig.  105,f  where  A  shows  field  of  main  pole  alone  and  B  and  C 

n    Auxiliary 
Pole 


N 


D 


N 


N 


N 


Brush 


Brush 


Brush 


FIG.  105. 


show  the  two  forms  produced  when  the  auxiliary  pole  is  used; 
in  B  its  current  is  in  the  same  direction  as  the  main-field  current 
and  in  C  it  flows  in  the  opposite  direction.  As  will  be  noticed 

*  For  curves  illustrating  this  point  see  Appendix,  Plate  27. 
t  For  curves  showing  experimentally  determined  field  forms  see  Plates 
24-26  of  the  Appendix. 


THE  ROTARY  CONVERTER 


177 


from  the  figures  considerable  change  may  occur  in  the  field  at  the 
point  of  commutation.  In  this  type  of  converter  there  is  some 
change  hi  the  wave  shape  of  the  A.C.  E.M.F.  when  the  field 
is  altered.  This  results  in  a  magnetizing  current  flowing  into 
the  machine  from  the  line  and  reduces  its  power  factor.  In  order 
to  keep  this  power  factor  as  high  as  possible  the  main  field  has 
to  be  changed  somewhat  as  the  auxiliary  field  is  changed,  thus 
tending  to  keep  the  C.E.M.F.  wave  of  the  same  form  and  mag- 
nitude as  that  of  the  impressed  E.M.F. 

An  idea  of  the  corrective  effect  of  the  magnetizing  current 
which  flows  into  the  A.C.  side  of  a  converter  when  there  is  a 
difference  in  wave  form  between  the  generated  and  impressed 
E.M.F.'s  may  be  obtained  by  reference  to  Fig.  106,  in  which  the 


-*,  f 

N 

s 

jff 

////////an  \  \ 

// 

v///nn\\\\ 

Distorted  E.M.F.—_JSine  E.M.F.  impressed 
generated    /Xl^v 


/[ 

'      V      current 
Resultant  E.M.F. 

FIG.  106. 

field  when  not  subject  to  armature  reaction  is  supposed  to  be  of 
such  distribution  as  to  give  the  generated  voltage  of  a  form  shown 
by  the  curve  so  marked,  while  the  impressed  E.M.F.  is  supposed 
to  be  a  sine  curve  shown  by  the  sine  curve  of  Fig.  106. 

From  a  to  b  the  coil  is  in  too  strong  a  field  and  the  C.E.M.F. 
is  greater  than  the  impressed  so  that  the "  magnetizing  "  current 
taken  from  the  line  during  the  time  a-b  will  be  a  demagnetizing 
current.  (It  was  seen  in  the  experiment  in  the  synchronous 
motor  that  an  overexcited  motor  draws  a  leading  current,  de- 
magnetizing the  motor  field,  and  an  underexcited  motor  a  lagging 
current,  magnetizing  the  motor  field  from  the  line.)  Hence, 
during  the  time  a-b  the  whole  field  embraced  by  the  coil  will  be 
subjected  to  a  demagnetizing  action.  At  time  B  this  magnetiz- 
ing current  falls  to  zero  value  because  the  normal  field  of  the 
rotary  is  of  the  proper  value  to  give  a  C.E.M.F.  equal  to  the 
line  E.M.F.  During  time  b-e  the  rotary  voltage  is  less  than 


178  ALTERNATING  CURRENTS 

the  impressed  E.M.F.  and  so  the  magnetizing  current  will  flow  in 
a  direction  opposite  to  what  it  had  during  the  time  a-b.  Hence, 
while  the  coil  travels  from  b  to  c  the  rotary  field  is  subjected  to 
the  magnetizing  action  of  the  armature  and  so  the  whole  field 
in  the  coil  becomes  stronger,  thereby  raising  the  value  of  the 
C. E.M.F.  as  it  should  be  raised  during  the  time  b-c. 

The  magnetizing  current  alternates  with  a  frequency  equal  to 
one-half  the  number  of  contacts  of  the  two  E.M.F.  waves.  Hence 
the  field  of  the  rotary  will  fluctuate  with  the  same  frequency. 
When  this  fluctuation  in  strength  of  the  rotary  field  is  examined 
more  closely  it  is  seen  that  it  really  produces  a  field  which  oscil- 
lates across  the  pole  face,  i.e.,  at  one  instant  the  field  is  strongest 
at  a,  and  then  at  6,  etc.  This  fluctuation  in  the  field  strength 
has  two  bad  effects.  It  dissipates  energy  in  the  pole  faces  by 
the  production  of  eddy  currents,  and  it  is  likely  to  cause  "  hunt- 
ing "  of  the  rotary.  As  explained  in  the  analysis  of  the  synchro- 
nous motor  this  hunting  may  be  much  lessened  by  making  the 
field  "  stiff er,"  usually  accomplished  by  placing  heavy  grids  of 
some  good  conductor  in  slots  in  the  pole  faces.  The  flux  in 
crossing  this  grid  sets  up  large  eddy  currents  in  the  bars  of  the 
grid,  which  currents  react  upon  the  field  to  retard  its  motion. 

In  operating  a  split-pole  converter  it  is  desirable  to  change 
the  current  in  the  auxiliary  pole  continuously  from  a  positive 

maximum  to  a  negative  maximum.  For 
this  purpose  a  special  rheostat  with  two 
contact  arms  is  used.  The  connections 
of  this  rheostat  are  designated  in  Fig. 
107  where  A  and  B  represent  the  mov- 
ing arms  which  travel  across  the  resist- 
F  107  ance  in  opposite  directions.  With  A  at 

a  the  auxiliary  field  gets  a  maximum 

current  in  one  direction;  when  A  and  B  come  together  in  the 
middle  the  field  has  no  excitation  and  then  as  A  moves  farther 
towards  b  the  current  reverses.  A  and  B  are  mounted  on  the 
same  shaft  but  on  oppositely  facing  dials,  so  that  by  the  same 
shaft  motion  one  is  moved  in  a  clockwise  direction  and  the  other 
travels  in  the  opposite  direction. 

The  compounding  in  this  type  of  rotary  might  be  made  more 
or  less  automatic  by  using  the  auxiliary  pole  to  carry  a  series  field, 
but  such  is  not  done  on  commercial  machines. 

Make  a  study  of  the  split-pole  rotary  and  try  different  methods 


THE  ROTARY  CONVERTER  179 

of  synchronizing;  notice  any  peculiarity  in  its  behavior  and 
explain  it.  Obtain  the  external  characteristic  of  the  rotary  when 
only  the  main  field  is  used;  adjust  the  main  field  before  syn- 
chronizing so  that  it  gives  the  proper  voltage  on  the  A.C.  side, 
and  keep  the  value  of  field-circuit  resistance  at  this  value  during 
the  run.  Read  amperes,  volts  and  watts  input,  and  amperes 
and  volts  output  and  field  current. 

With  no  load  on  the  machine  and  maintaining  constant  im- 
pressed voltage,  take  a  series  of  readings  to  show  how  the  voltage 
ratio  may  be  changed  by  means  of  the  auxiliary  pole.  Begin 
with  maximum  negative  current  in  the  auxiliary  poles  and  adjust 
the  main  field  to  give  maximum  power  factor  (i.e.,  to  make  the 
C.E.M.F.  wave  as  nearly  as  possible  similar  to  the  impressed 
E.M.F.  wave).  Read  amperes,  volts,  watts  and  power  factor 
(with  power-factor  meter)  on  A.C.  side,  volts  on  D.C.  side  and 
both  field  currents.  Note,  by  sparking  at  brush  contacts,  if  the 
field  distortion  produces  shifting  of  the  commutation  plane. 
Get  about  eight  readings  in  the  maximum  range  of  the  auxiliary- 
pole  field  current. 

Make  a  similar  run  with  full  load  on  the  converter,  making 
same  adjustments  as  before  and  reading  the  same  quantities  and 
also  D.C.  load  current,  which  is  to  be  maintained  constant  at  rated 
full-load  value.  Try  one  run  as  above  with  the  commutating 
poles  and  one  without  them  and  note  especially  their  effect  upon 
commutation. 

Running  the  machine  from  the  D.C.  end  with  constant  im- 
pressed E.M.F.  take  ondograph  curves  of  the  E.M.F.  wave  forms 
between  the  different  taps;  by  means  of  a  search  coil  and  ondo- 
graph get  proper  curves  to  show  the  field  distribution  under 
three  conditions  of  the  auxiliary-pole  excitation  (zero  and  maxi- 
mum in  both  directions)  and  obtain  under  the  same  conditions 
the  A.C.  voltage  generated  between  180°  taps  and  120°  taps  on 
the  A.C.  winding. 

Plot  curves  of  external  characteristic,  efficiency,  power  factor, 
and  field  current,  from  the  results  obtained  in  the  run  when  main 
field  only  was  used;  use  load  current  as  abscissae.  Plot  on  an- 
other sheet  curves  between  voltage  ratios  and  auxiliary  field  cur- 
rent, and  on  a  third  sheet  curves  between  main  field  and  auxiliary 
field  current,  using  auxiliary  field  as  abscissas.  On  these  two 
sheets  plot  curves  for  both  the  no-load  and  full-load  runs. 

There  are  in  use  two  other  methods  for  compounding  rotary 


180 


ALTERNATING  CURRENTS 


converters  besides  the  two  illustrated  in  Experiments  31  and  32. 
One  method  uses  an  induction  regulator  in  series  with  the  A.C. 
supply  to  the  rotary,  and  the  other  method  employs  a  synchro- 
nous booster  on  the  same  shaft  as  the  rotary  armature. 

A  single-phase   induction  regulator  uses  the  principle   of   a 
variable  ratio  transformer.     One  coil  is  connected  in  shunt  with 

the  line  supplying  the  rotary 
with  A.C.  power  and  the  other 
coil,  in  series  with  the  line  sup- 
plying power  to  the  rotary, 
consisting  of  fewer  turns  than 
the  first  (the  number  of  turns 
depending  upon  the  amount 
of  compounding  desired)  is 
placed  at  right  angles  to  the 
first.  A  movable  iron  core 
changes  both  the  direction 
and  amount  of  flux  produced 

by  the  first  coil  which  cuts  the 

JTIG   io8.  second,  the  iron  core  being  in 

position  D  (Fig.  108)  to  assist 

the  impressed  E.M.F.  and  in  position  Df  to  crush  the  impressed 
E.M.F.  The  two  coils  A 
and  B  are  wound  in  slots  in 
the  laminated  yoke  C,  and 
the  movable  core  D  is  ma- 
nipulated by  means  of  a 
hand- wheel  and  worm  gear. 
The  polyphase  induction 
regulator  is  essentially  an 
induction  motor  with  a 
wound  rotor.  The  rotor 
cannot  revolve  but  may  be 
moved  (automatically  or 
not,  as  desired)  through  an 
arc  of  about  180  electrical 
degrees.  The  stator  winding  is  connected  across  the  supply  line 
and  the  rotor  winding  is  put  in  series  with  the  power  supply  of 
the  rotary  (or  vice-versa).  The  magnitude  of  the  voltage  gen- 
erated by  the  rotor  coils  is  constant  (as  a  polyphase  magnetic 
field  is  practically  constant  in  strength)  but  the  phase  in  which  this 


THE  ROTARY  CONVERTER  181 

rotor  voltage  is  combined  with  the  line  voltage  may  be  altered 
by  turning  the  rotor  in  different  angular  positions.  In  Fig.  109, 
OA,  OB  and  OC  represent  the  supply  voltages  and  A  A',  BE' 
and  CC',  the  voltage  generated  in  the  rotor  coils.  Accordingly 
as  these  E.M.F.'s  are  added  to  the  line  E.M.F.  in  phase,  180°  out 
of  phase,  90°  out  of  phase,  etc.,  the  rotary  will  be  supplied  with 
voltage  OA',  OA"  OA'",  etc. 

In  the  synchronous-booster  system  a  machine  of  the  same 
number  of  poles  as  the  rotary  is  placed  on  the  same  shaft  with 
the  rotary;  the  armature  windings  of  this  machine  connect  to 
slip  rings  on  one  side  and  on  the  other  connect  to  the  A.C.  taps 
of  the  rotary.  The  field  of  this  synchronous  booster  is  adjustable 
and  the  supply  voltage  to  the  rotary  may  be  changed  by  varying 
this  field,  the  maximum  range  being  from  full  negative  field  of 
the  booster  to  full  positive  field.  If  the  rotary  is  to  be  com- 
pounded automatically  the  booster  field  may  be  wound  in  two 
coils,  one  carrying  a  field  current  which  crushes  the  line  voltage 
and  remains  constant,  the  other  coil  will  be  in  series  with  the 
rotary  load  and  has,  at  full  load  on  rotary,  twice  as  many 
ampere  turns  as  the  other  coil.  By  this  means  the  effect  of 
the  booster  is  automatically  changed  from  a  negative  E.M.F.  to 
a  positive  E.M.F.  as  the  rotary  load  increases. 

At  no  compounding  the  booster  is  more  efficient  than  the  in- 
duction regulator  because  it  has  no  core  losses,  while  the  regu- 
lator core  losses  are  independent  of  the  amount  of  compounding 
obtained  by  its  use. 

If  time  permits  make  a  study  of  the  induction  regulator, 
polyphase  and  single  phase,  also  run  a  test  on  a  booster  com- 
pounded rotary. 


EXPERIMENT   XXXIII. 

STUDY  OF  THE  INDUCTION  MOTOR;  OBTAINING  ITS  CHARACTER- 
ISTICS BY  LOADING  WITH  PRONY  BRAKE  OR  GENERATOR. 

THE  production  of  the  rotating  field  in  an  induction  motor  by 
polyphase  currents  and  windings  has  been  so  thoroughly  dis- 
cussed in  various  text-books  that  it  will  here  be  taken  as  under- 
stood.* Although  the  field  in  such  motors  is  not  exactly  constant 
it  is  so  nearly  of  uniform  strength  that  for  test  purposes  it  may 
be  considered  so. 

In  the  squirrel-cage  rotor  the  conductors  are  short-circuited 
through  end  rings  mounted  on  the  rotor;  in  the  wound  rotor 
the  ends  of  the  rotor  windings  are  connected  to  slip  rings,  which 
rings  may  be  either  short-circuited  or  connected  together 
through  suitable  resistances.  In  the  squirrel-cage  rotor  the 
resistance  of  the  paths  may  be  made  very  low  as  the  winding 
generally  consists  of  only  one  bar  per  slot,  but  in  the  wound  rotor 
the  resistance  of  the  paths  cannot  be  made  extremely  low  as  the 
brushes  and  brush  contacts  offer  considerable  resistance,  however 
small  the  resistance  of  the  winding  may  be. 

The  currents  in  the  rotor  and  stator  hold  the  same  relations  to 
one  another  as  in  a  static  transformer  and  it  is  therefore  evident 
that  if  but  few  turns  are  used  in  the  rotor  its  current  must  be 
correspondingly  large;  as  such  large  currents  would  demand 
heavy  rings  and  brush  rigging  and  would  dissipate  a  lot  of 
energy  in  the  form  of  heat  at  contact  points,  it  is  customary  to 
put  in  the  rotor  winding  a  number  of  conductors  much  greater 
than  in  the  case  of  squirrel-cage  rotors.  In  two  sample  motors, 
e.g.,  the  rotor  has  one-third  and  one-fourth  as  many  turns  as  the 
stator  winding.  The  rotor  and  stator  are  not  necessarily  wound 
with  the  same  number  of  phases. 

The  torque  of  a  polyphase  induction  motor  is  given  by  the 
equation:  JV22eV2s 

co(r22  +  s2z22)' 

*  See  Crocker  and  Arendt,  "  Electric  Motors,"  page  180. 
t  See  Crocker  and  Arendt,  "  Electric  Motors,"  page  165,  et  seq. 

182 


THE  POLYPHASE  INDUCTION   MOTOR 


183 


Where         N2  =  number  of  turns  per  secondary  circuit. 

e  =  induced  volts  per  turn  of  secondary  circuit  when 

rz  =  resistance  per  secondary  circuit. 

x2  =  reactance  per  secondary  circuit  when  s  =  I. 

s  =  rotor  slip. 

co  =  speed  of  rotating  field. 

This  torque  evidently  depends  upon  r2  and  it  is  found  to  be 
a  maximum  when  r2  =  sx2.  This  condition  is  obtained  by  dif- 
ferentiating the  torque  equation  with  respect  to  r2  and  placing 
its  coefficient  equal  to  zero. 

The  physical  significance  of  this  condition  for  maximum 
torque  may  be  obtained  by  analyzing  the  relation  between  the 
current  and  field  which 
produce  the  torque.  The 
field  of  an  induction  motor 
is  generally  such  that  its 
space  distribution  around 
the  periphery  of  the  rotor 
may  be  represented  by 
the  equation  <£  =  <f>m  cos  a, 
where  a  is  the  angle  (in 
electrical  degrees)  between 
the  point  considered  and 
the  axis  of  the  field,  as 
shown  in  Fig.  110. 

Although  the  following 
analysis   may  be  readily  FIG.  110. 

carried  out  for  any  value 

of  slip,  it  becomes  simpler  when  it  is  assumed  that  s  =  1,  and  such 
value  of  s  is  here  used.  The  rotor  being  stationary  and  the  field 
rotating  with  a  velocity  of  co  in  the  direction  shown,  there  will 
be  produced  in  conductor  A  an  E.M.F.  which  has  a  frequency 

of  ^— ,  and  has  its  maximum  value  when  the  conductor  occupies 

Z  7T 

the  most  dense  part  of  the  field,  as  shown  in  Fig.  110  at  A. 

If  A  is  part  of  a  rotor  coil  of  very  high  resistance  the  current 
in  A  will  be  in  phase  with  the  generated  E.M.F.  and  will  have  its 
maximum  value  when  the  conductor  occupies  the  position  shown 
at  A.  If  the  reactance  of  the  coil  is  very  high  the  current  will 
reach  its  maximum  value  when  the  E.M.F.  has  fallen  to  zero;  for 


184  ALTERNATING  CURRENTS 

maximum  current  value  the  conductor  will  have  the  position 
shown  by  A",  as  here  the  conductor  occupies  a  field  of  zero 
intensity,  where  the  induced  E.M.F.  is  evidently  zero.  In 
Fig.  110  it  is  assumed  that  the  rotor  has  turned  backward  90°. 
(We  assumed  the  rotor  to  be  stationary  and  the  field  to  move,  but 
it  is  easier  in  the  diagram  to  show  the  relative  motion  by  moving 
the  conductor  from  A  to  A".)  But  whenever  either  the  resist- 
ance of  the  coil  or  the  reactance  is  very  large  no  torque  is  pro- 
duced, as  may  be  seen  from  Equation  (2). 

If  i2  =  value  of  current  in  conductor  when  it  is  situated  in  a 
field  of  density  0,  then 

torque  =  K<j>i2,  (2) 

where  K  is  a  constant  depending  upon  length  of  conductor, 
diameter  of  rotor  and  units  in  which  /2  and  0  are  measured. 

If  Em  =  maximum  value  of  E.M.F.  induced  in  coil. 

Im  =  maximum  value  of  iV 

12  =  Im  G0s(cot  -  0)  =  Em  _    COS  ((at  -  6) 


=  —  cos  (wt  -  0)  sin  B, 

SX2 

in  which  t  =  o  when  conductor  is  in  magnetic  axis  (as  at  A)  and 


where  sx2  and  r2  are  the  rotor  reactance  and  resistance.  Sup- 
pose that  at  the  instant  considered  the  conductor  is  as  shown 
at  A'.  Then 

torque  =  T  =  KIm<t>m  cos  /3  cos  ((at  —  0), 
but  as 

|8  =  (at,    T  =  KIm(j)m  cos  at  COS  (wt  -  6). 

The  torque  produced  thus  by  a  single  conductor  is  a  double- 
frequency  quantity  but  it  can  be  shown  that  the  resultant  torque 
of  all  the  conductors  on  the  rotor  is  a  constant  quantity,  the 
value  of  which  depends  upon  the  maximum  value  of  the  torque 
produced  by  one  conductor. 

Now  the  maximum  value  of  the  torque  produced  by  any  con- 
ductor depends  upon  the  maximum  value  of  the  current  in  the 
conductor  and  the  intensity  of  the  field  in  which  the  conductor 
is  situated  when  the  current  reaches  its  maximum  value. 


THE  POLYPHASE  INDUCTION  MOTOR  185 

The  maximum  current  in  a  coil 

Em  E 


The  strength  of  field  in  which  the  conductor  lies  when  this 
maximum  current  occurs  is  </>  =  <f>m  cos  j(3,  but  it  will  be  noticed 
that  the  time  phase  of  the  current  =  space  phase  of  field,  so  that 
/3  =  0,  hence:  torque  when  maximum  current  occurs 

T  =  —  0m  sin  ]8  cos  j3. 

sxz 

To  get  condition  for  maximum  torque,  we  put 


or  cos2  j3  =  sin2  /3  which  means  that  /3  =  45°. 

Hence,  for  a  given  inductance  of  the  rotor  coil  the  torque  will 
be  greatest  when  the  resistance  of  the  coil  is  so  adjusted  that  the 
angle  of  lag  of  current  hi  the  coil  is  45°.  But  this  occurs  when 
sx2  =  r2. 

The  above  discussion  is  carried  out  in  the  assumption  that 
s  =  1,  as  we  supposed  the  rotor  stationary.  The  same  result  will, 
however,  be  obtained  whatever  value  is  assumed  for  the  slip. 

To  get  a  high  starting  torque  the  resistance  r2  should,  there- 
fore, be  made  equal  to  the  standstill  reactance  of  the  rotor; 
but  such  a  high  value  of  resistance  will  give  a  very  small  torque 
near  synchronous  speed  because  here  sx%  is  small  and  r2  should 
be  correspondingly  small.  Of  two  motors  having  the  same  stray- 
power  losses,  the  one  having  the  smaller  slip  for  a  given  load 
will  have  the  higher  efficiency.  It  may  be  shown  that  if  the 
stray-power  and  primary  I2R  loss  are  neglected  the  efficiency  of 
a  motor  =  (1  —  s).*  Therefore  an  induction  motor  should  give 
its  rated  load  with  as  small  a  value  of  slip  as  is  possible. 

Using  the  same  symbols  as  before 


and  as  Em  is  fixed,  for  a  certain  value  of  Im  (as  e.g.,  that  required 
for  rated  output)  as  we  have 

Im2 

•2  =  K,  a  constant,  independent  of  s. 


McAllister,  "  Alternating  Current  Motors,"  page  21. 


186 


ALTERNATING  CURRENTS 


Putting 
we  have 


/    2 


r2 


rta 


or   s2  = 


d 

^ 

O   uJ 

/ 

\ 

5? 

j 

\ 

t^ 

/ 

\ 

be  9  ft 

/ 

S 

"**. 

E 

' 

••  ^ 

3  10 

*  ^ 

CQAU 

*  •*.  ^ 

1 

h 

(I 

or 

v 

O' 

is 

m 

OP 

0 

o 

L  ns 

1 

I 

I 

| 

1.5 


/  ^2 

which  gives  s  =  ar2,  where  a  =  y ^=^,  a  constant. 

From  this  it  is  evident  that  if  it  is  desired  to  keep  the  slip  small, 
TZ  must  be  kept  small.  Hence  the  conditions  for  good  starting 
torque  and  high  running  efficiency  conflict  with  one  another. 

The  problem  is  solved  commercially  by  having  a  wound  rotor, 
generally  three  phase.  The  rotor  slip  rings  are  connected  through 
a  three-phase  resistance  and  a  controlling  switch  by  which  the 

amount  of  external  re- 
sistance in  each  phase 
can  be  cut  down  in 
steps  from  a  maximum 
at  starting  to  zero  at 
normal  speed.  For 
small  motors  the  resis- 
tance is  sometimes 
located  in  the  rotor 
spider  and  a  lever  with 
a  sliding  sleeve  on  the 

shaft  takes  the  place  of  the  control  switch  used  with  larger 
motors. 

The  variation  of  starting  torque  with  rotor  circuit-resistance 
is  given  in  Fig.  Ill,  in  which  case  the  maximum  starting  torque 
occurs  when  the  resist- 
ance of  the  rotor  circuit 
is  0.4  ohm.  Hence,  if  the 
rotor  circuit  itself  has  a 
resistance  of  0.1  ohm, 
the  proper  external  re- 
sistance to  add  for  maxi- 
mum starting  torque  is 
0.3  ohm.  For  a  three- 
phase  rotor  the  circuit  would  be  as  shown  in  Fig.  112,  in  which 
the  three-phase  external  resistance  and  controlling  switch  are 
shown.  For  starting,  the  switch  position  is  shown  by  the  full 


.5  ohm  1.0 

Rotor  circuit  resistance 

FIG.  111. 


External  Resistance 


THE  POLYPHASE  INDUCTION  MOTOR 


187 


lines  in  the  diagram,  while  for  normal  running  conditions  the 
switch  would  be  in  the  position  shown  by  the  dotted  lines. 

Besides  increasing  the  starting  torque  the  introduction  of  extra 
resistance  into  the  rotor  circuit  has  the  effect  of  cutting  down  the 
starting  current  of  the  motor.  The  motor  is  essentially  a  short- 
circuited  transformer  and  even  though  the  reactance  of  the 
windings  is  high  (due  to  leakage  flux)  still  the  current  taken  at 
standstill  may  be  several  times  full-load  current  unless  the  exter- 
nal resistance  is  used.  The  starting-current  curve  of  the  motor 
having  external  resistance  is  shown  in  Fig.  113.  The  curve 
ABCDE  shows  the  variations  in  starting  current  as  the  various 


A1 

1! 

; 

*•*, 

^-* 

^^ 

^* 

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, 

X1 

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«^ 

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S 

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\\ 

<-^ 

^ 

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x 

N 

> 

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^  j 

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/,- 

•*- 

, 

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^ 

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\ 

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"7 

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s 

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£ 

\ 

/ 

\ 

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s 

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^ 

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5 

-V 

If 

o 

,-: 
-~. 

^ 

.- 

— 

—  »« 

._ 

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. 

"t 

' 

\ 

*^ 

\ 

S 

^ 

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~^* 

•*• 

•> 

< 

„ 

>, 

y 

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_,. 

^ 

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^ 

r\ 

^ 

•^ 

J 

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>^ 

i 

\ 

\  1  1  1 

,- 

^. 

-* 

^ 

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t  \  1  1 

[-" 

_  H 

*• 

^ 

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/.  \ 

SB 

. 

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„ 

--• 

x 

1 

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t 

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3 

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k 

,M 

,- 

— 

NS 

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\  UI 

__ 

—  - 

\TV\U 

-- 

^\.i 

4 

0           10           20          30          40          50          60          70          80          90         100 
Per  cent  pf  Synchronous  Speed 

FIG.  113. 

steps  of  extra  resistance  are  cut  out  with  increasing  speed  and 
the  curve  A'B'C'D'E'  gives  the  corresponding  speed-torque 
relation.  If  the  control  switch  is  properly  used  the  motor  will 
operate  at  nearly  maximum  torque  (=  OA')  during  the  whole 
time  of  speeding  up.  The  current  OA  is  generally  somewhat 
more  than  full-load  current. 

A  small  motor,  equipped  with  squirrel-cage  rotor,  gives  small 
starting  torque  and  excessive  starting  current. 

Curves,  similar  to  those  of  Fig.  113,  are  given  hi  Fig.  114  for 
a  squirrel-cage  motor  thrown  directly  on  the  line.  From  a  com- 
parison of  these  two  sets  of  curves  the  advantage  of  the  wound 
rotor  over  the  squirrel-cage  rotor  is  at  once  apparent. 

Many  times  a  motor  equipped  with  squirrel-cage  rotor  is  fitted 
with  a  double-throw  switch  and  is  first  thrown  on  to  low-voltage 


188 


ALTERNATING  CURRENTS 


taps  of  the  supply  transformers  and  not  connected  directly  to 
the  line  of  normal  voltage  until  approaching  synchronous  speed. 
The  advantage  of  low-voltage  taps  is  the  decrease  in  starting 
current.  The  starting  torque  is  also  decreased  by  this  method 
of  starting. 

The  direction  of  rotation  of  a  three-phase  motor  may  be 
reversed  by  interchanging  two  of  the  supply  wires  and  of  the 
two-phase  motor  by  interchanging  the  wires  of  either  phase. 

The  effect  of  increasing  the  impressed  voltage  of  an  induction 
motor  is  to  increase  its  speed  for  a  given  torque,  increase  of 
"  pull-out  torque,"  increased  magnetizing  current  and  iron  loss 
and  slight  increase  in  efficiency.  The  increase  in  magnetizing 


M)  40  50  60  70 

Per  cent  Synchronous  Speed 
FIG.  114. 


100 


current  becomes  very  rapid  for  an  increase  of  voltage  of  more  than 
about  20  per  cent  above  rating  and  results  in  poor  power  factor. 

A  variation  in  the  frequency  of  the  supply  gives  results  similar 
to  those  obtained  for  the  transformer,  a  decrease  in  frequency 
causing  larger  magnetizing  current,  greater  core  loss,  etc. 

Under  normal  conditions  the  slip  of  an  induction  motor  having 
no  extra  resistance  in  its  rotor  circuit  is  small  and  the  speed 
cannot  be  accurately  enough  obtained  by  the  ordinary  speed 
counter.  Several  methods  have  been  devised,  however,  which 
permit  very  exact  determinations  of  the  speed.  They  all  work 
on  the  idea  of  obtaining  the  slip,  from  which  the  actual  speed  can 
easily  be  obtained. 

The  stroboscopic  method  employs  an  A.C.  arc  fed  from  the 
supply  line  of  the  motor  and  a  disc  mounted  on  the  end  of  the  rotor 


THE  POLYPHASE  INDUCTION   MOTOR 


189 


shaft.  This  disc  is  painted  alternately  black  and  white  in  sectors, 
there  being  as  many  white  sectors  as  there  are  poles  on  the  motor 
(or  some  submultiple  of  the  number  of  poles  on  the  motor). 

Suppose  a  two-pole  motor  is  considered;  the  disc  will  be  painted 
in  quadrants  one  black  and  the  next  white  as  in  Fig.  115.  Sup- 
pose the  rotor  is  turning  at  synchronous  speed  and  that  the  rotor 
is  in  the  position  shown  when  maximum  E.M.F.  occurs,  which  will 
be  the  time  when  the  A.C.  arc  is  most  brilliantly  illuminating 
the  disc.  As  sector  a  moves  away  from  the  upper  position  the 
arc  dies  down  and  does  not  again  have  its  maximum  illuminating 
power  for  one-half  cycle,  hi  which  time,  if  the  rotor  is  turning 
synchronously,  a  will  be  in  the  lower  position  on  the  disc  and 


Reflector 


FIG.  115. 

white  sector  c  will  be  occupying  the  position  which  a  occupied 
one-half  period  previously.  The  eye  is  unable  to  distinguish  the 
difference  between  sectors  a  and  c  and  so  gives  the  impression 
that  sector  a  is  stationary  in  space. 

When  such  a  disc  is  turning  at  synchronous  speed  the  disc, 
therefore,  appears  stationary;  but  if  the  rotor  has  some  slip, 
sector  c  will  occupy  some  position  lower  than  the  upper  one  at 
the  end  of  one-half  period,  while  sector  a  will  occupy  a  position  on 
the  lower  side  of  the  disc  but  not  so  far  advanced  as  that  occupied 
by  c  one-half  period  previously.  The  eye  interprets  this  phenom- 
ena by  giving  the  observer  the  impression  that  sector  a  (and  of 
course  all  the  other  sectors)  are  moving  slowly  in  a  direction 
opposite  to  the  motion  of  the  rotor.  The  number  of  apparent 
revolutions  of  the  disc  per  minute  divided  by  synchronous  speed 
gives  the  per  cent  slip. 


190  ALTERNATING  CURRENTS 

Another  convenient  method,  for  measuring  the  slip,  consists  of 
lighting  an  incandescent  lamp  from  the  motor's  supply  circuit 
through  an  insulating  disc  having  a  conducting  strip  on  part  of 
its  periphery.  The  disc  and  brush  may  be  mounted  on  the 
spindle  of  a  speed  counter.  Connections  for  this  scheme  are 

shown  in  Fig.  116.     If 
the  disc  is  turning   at 
PfoVreMotoPrPl7    synchronous  speed  the 
condenser   will    receive 
the     same    charge     at 
Metal  /OPN '  every  contact  so  that  the 

Strip-riT)     )  Fiber  disc  on  spindle  ,  .„  ,  .,, 

1  of  revolution  counter  lamp  Will  burn  With  uni- 

FlG  116  form  brilliancy,   but  if 

the  disc  is  running  at  less 

than  synchronous  speed  the  condenser  receives  a  varying  charge 
according  to  the  value  of  the  line  voltage  at  the  time  of  contact. 
The  lamp  will  give  a  bright  period  for  every  time  the  rotor  slips 
an  alternation,  i.e.,  for  every  180°  (electrical)  of  slip.  Hence,  the 
number  of  bright  periods  per  minute  divided  by  the  number  of 
poles  on  the  motor  gives  the  slip  in  r.p.m.  jmd  the  slip  in  per  cent 
is  obtained  by  dividing  this  number  by  the  (revolutions  indicated 
on  the  counter  -+-  the  number  obtained  as  slip). 

If  the  slip  is  so  great  that  the  number  of  flickers  cannot  be 
accurately  counted,  the  connections  as  given  in  Fig.  116  may  be 
modified  as  shown  in  Fig. 
117.  By  connecting  theD.C. 

,.        .     J      .  *  — *  A  A  Power  Supply 

line  in  series  with  the  A.C.  v_ j  Y       for  Motor 


line,  the  number  of  flickers 

per  second  will  be  reduced  to 

one-half  the  previous  value. 

The  effect  of  the  D.C.  line  hi  FIG.  117. 

series  with  the   A.C.   is  to 

make  all  of  the  alternations  positive  (or  negative)  ;  of  course  the 

lamp  used  must  be  able  to  stand  220  volts  instead  of  110  volts  as 

for  connections  of  Fig.  116.     With  connections  as  made  in  Fig.  117 

the  formula  for  obtaining  slip  becomes 

number  of  flickers 


01. 
Slip 


—  :  --  5  —  ;  -   —  T— 
pairs  of  poles  on  motor 


Many  modifications  of  this  method  will  occur  to  the  student,  e.g., 
counting  the  beats  in  a  telephone  receiver  properly  connected. 


THE  POLYPHASE  INDUCTION   MOTOR  191 

There  is  on  the  market  a  slip-meter,  from  the  indications  of 
which,  by  use  of  a  table  sent  with  the  instrument,  the  per  cent 
slip  may  be  directly  obtained. 

Study  the  construction  of  the  three  types  of  motors  discussed 
above.  Try  the  effect  upon  the  rotation  of  reversing  one  of  the 
phases ;  of  reversing  two  of  them. 

Perform  a  load  run  on  a  polyphase  motor,  using  a  prony  brake 
or  generator  for  load.  Impress  normal  voltage  and  frequency 
and  keep  these  quantities  constant  during  the  run.  Measure  the 
current,  volts,  watts  input  and  torque.  Obtain  speed  by  using 
one  of  the  above-mentioned  methods  for  obtaining  slip  and  calcu- 
late synchronous  speed  from  frequency  and  number  of  poles. 
If  the  number  of  poles  cannot  be  ascertained  by  examination  run 
the  motor  light,  under  which  condition  it  will  run  at  a  speed 
within  a  fraction  of  one  per  cent  of  synchronism,  from  which  the 
number  of  poles,  and  hence  synchronous  speed,  can  be  computed. 
A  very  little  practice  with  induction  motors  is  needed  to  tell 
immediately  the  number  of  poles;  e.g.,  a  motor  supplied  with 
60-cycle  power  running  at  875  r.p.m.  could  not  be  other  than  an 
8-pole  motor.  (If  external  resistance  is  used  in  the  rotor  circuit 
this  method  of  reasoning  will  not  hold,  as  it  might  be  a  6-pole 
motor  with  a  proper  resistance  in  the  rotor  to  cut  the  speed  down 
from  its  normal  value,  which  would  be  between  1150  r.p.m.  and 
1200  r.p.m.) 

Take  readings  with  the  motor  running  light,  also  at  |,  i,  \,  \ , 
full  and  1 J  rated  load.  For  setting  the  brake  to  its  proper  value 
for  these  loads,  the  torque  for  the  various  loads  may  be  calculated 
upon  the  assumption  that  the  motor  will  maintain  synchronous 
speed.  Then  adjust  the  brake  to  get  approximately  the  value 
so  determined. 

Plot  curves  of  speed,  efficiency,  power  factor,  current  (equiva- 
lent single-phase)  and  torque,  with  H.P.  output  as  abscissa  for 
all  curves. 

Note.  —  If  a  three-phase  motor  is  used,  two  wattmeters  will  be  used  and 
at  light  load  one  of  them  is  very  likely  to  read  negatively.  See  experiment  on 
"  Measurement  of  three-phase  power." 


EXPERIMENT  XXXIV. 

PREDICTION  OF  INDUCTION-MOTOR  CHARACTERISTICS  BY  THE 
METHOD  OF  THE  CIRCLE  DIAGRAM. 

ALL  of  the  characteristics  of  an  induction  motor  can  be  pre- 
determined from  the  "  circle  diagram,  "  for  the  construction  of 
which  it  is  necessary  to  take  only  two  sets  of  readings  from  the 
motor.  The  application  of  this  circle  diagram  depends  upon 
the  following  fact;  as  the  load  upon  an  induction  motor  in- 
creases its  rotor  circuit  may  be  accurately  represented  by  a 
resistance  and  inductance,  in  series  with  each  other,  connected 
to  a  constant-potential  line,  the  resistance  having  different  values 
corresponding  to  different  loads  upon  the  motor,  and  the  induct- 
ance remaining  constant.  The  locus  of  current  for  such  a 
circuit  is  a  semicircle,  as  was  proved  in  Experiment  9.  As  the 
induction  motor  is  essentially  a  transformer  in  its  current  and 
E.M.F.  relations,  any  current  in  the  rotor  must  have  an  equal 
and  opposite  current  in  the  stator.  The  stator  current  will 
be  represented,  therefore,  by  the  rotor  current  plus  whatever 
current  flows  in  the  stator  circuit  when  the  rotor  current  is 
zero. 

Before  the  use  of  the  circle  diagram  is  justified  it  must  be 
shown  that  the  effect  of  putting  mechanical  load  on  the  motor 
has  the  same  effect  upon  the  stator  and  rotor  currents  as  is 
produced  upon  the  secondary  and  primary  currents  of  a  trans- 
former when  the  secondary  resistance  is  varied. 

It  has  been  shown  previously  that  for  any  definite  value  of  rotor 
current  we  have  s  =  arz.  Now  when  s  =  I  the  motor  can  be 
doing  no  mechanical  work  and  so  all  the  energy  fed  to  the  rotor 
must  be  used  up  as  PR  loss,  as  it  is  in  a  static  transformer  feeding 
lamps;  but  for  any  value  whatsoever  of  /2,  r2  can  be  so  chosen 
that  «  =  1,  ((a)  involves  the  values  of  1 "2  considered).  Hence 
the  current  relations  in  any  induction  motor  can  be  accurately 
represented  by  considering  it  as  a  static  transformer  the  resist- 
ance of  whose  secondary  circuit  can  be  altered  to  correspond  to 
the  load  on  the  motor. 

192 


THE  POLYPHASE  INDUCTION   MOTOR 


193 


Load 


J£l      ri  X2      r2 

1  -  nnrsAA  —  i  —  •w-vs,  — 


2 


FIG.  118. 


As  is  usually  done,  the  motor  will  be  considered  a  transformer 
with  ratio  of  1 :  1  because  the  different  quantities  are  then  more 
easily  compared.  The  circuit  of  the  induction  motor  may  then 
be  represented  by  the  upper  diagram  in  Fig.  118,  which  is  really 
the  same  as  the  lower  figure.  The  TQ  and  XQ  paths  in  the  lower 
figure  are  of  such  values  that  their  admittances  permit  the  pass- 
age of  two  currents  of  such  value 
that  they  represent  accurately  the 
power  component  and  magnetizing 
component  of  the  primary  current 
in  the  upper  circuit  when  the  sec- 
ondary is  open.  The  combined 
current  of  the  paths  XQ  and  r0  repre- 
sents the  "  running  light  "  current 
of  the  induction  motor.  With  the 
circuit  depicted  in  Fig.  118,  this 

current  will  decrease  as  the  value  of  R  is  changed,  owing  to  the  drop 
through  ri  and  XL  This  change  in  "  running  light  "  current  actu- 
ally does  occur  when  the  motor  is  loaded  but  the  change  is  only 
a  few  per  cent.  As  this  "  running  light  "  current  is  itself  not 

large  compared  with  full-load  cur- 
rent of  the  motor,  the  error  intro- 
duced, by  considering  the  "running 
light  "  current  (/<>)  to  be  independ- 
ent of  load,  is  small,  and,  as  it 
simplifies  the  problem,  it  is  consid- 
ered constant  when  predicting  the 

behavior  of  an  induction  motor.  The  motor  circuit  is  now 
represented  by  Fig.  119.  In  such  a  circuit 

E 


FIG.  119. 


and  /  =  /i  +  /o  (vector  addition) . 

In  the  above  equation  for  /i,  the  only  variable  is  R,  hence  the 
circuit  is  equivalent  to  that  of  Experiment  9,  and  the  locus  of  I\ 


will  be  a  semicircle  of  diameter  = 


E 


;  the  locus  of  /  will  be 


the  same  semicircle  but  /  will  be  measured  from  a  different  origin. 
All  of  the  quantities  discussed  here  are  shown  in  their  proper 
relations  in  the  diagram  shown  in  Fig.  120. 


194 


ALTERNATING  CURRENTS 


The  measurements  necessary  for  the  constructing  of  this  dia- 
gram and  the  method  in  which  they  are  used  are  as  follows: 
With  normal  voltage  and  frequency,  the  input  in  volt-amperes 
and  watts  is  read  with  no  load  on  the  motor.  This  is  the  run- 
ning light  current  and  by  means  of  the  calculated  power  factor 
it  may  be  separated  into  its  energy  component  and  magnetizing 
component.  These  quantities  reduced  to  equivalent  single- 
phase  values  are  shown  in  Fig.  120,  OZ  is  the  running  light  cur- 
rent, OP  and  OQ  are 
its  magnetizing  and 
energy  components, 
respectively. 

The  next  test  to 
make  is  the  "  locked- 
saturation  "  curve. 
This  curve  is  obtained 
by  clamping  the  short- 
circuited  rotor  so  that 
it  cannot  move,  and 
impressing  various 
FIG.  120.  voltages  upon  the 

stator  and  reading  the 

corresponding  input  in  watts  and  amperes.  A  low  voltage  is 
impressed  at  first  and  this  is  increased  until  about  300  per  cent 
rated  current  is  flowing.  About  six  readings  distributed  through- 
out this  range  will  be  sufficient.  In  reading  the  higher  values  of 
current  the  work  must  be  done  very  rapidly  and  the  current  need 
not  be  brought  up  to  its  high  value  until  every  man  who  is 
engaged  in  the  test  is  ready  to  read.  As  soon  as  a  reading  has 
been  taken  the  rotor  should  be  allowed  to  revolve  for  a  few 
minutes  to  give  it  a  chance  to  cool  off. 

When  the  "  locked-saturation  "  curve  is  plotted  it  will  be 
found  to  be  practically  a  straight  line,  which  means  that  the 
impedance  of  the  windings  is  a  constant.  This  is  to  be  expected 
because  normally  the  iron  in  the  motor  is  worked  at  such  a  low 
density  that  even  300  per  cent  rated  current  may  not  saturate 
the  path  of  the  leakage  lines.  As  soon  as  the  teeth  become 
saturated  then  the  impedance  will  change. 

The  results  obtained  from  this  test  are  plotted  as  in  Fig.  121 
and  the  curve  extrapolated  to  get  current  with  normal  voltage 
impressed  (OA).  In  the  test  probably  not  more  than  50  per 


THE  POLYPHASE  INDUCTION   MOTOR 


195 


cent  normal  voltage  can  be  impressed  without  danger  of  burning 
the  motor.  From  the  value  of  watts  and  current  input  calculate 
the  effective  resistance  of  the  motor  for  the  different  readings. 
This  should  be  nearly  constant.  Use  the  average  value  of  this 
resistance  and  calculate  what  will  be  the  PR  loss  for  current  OC. 
So  the  power  factor  of  the  "  locked-saturation  "  input  at  normal 
voltage  can  be  calculated  and  so  the  current  OC  may  be  separated 
into  its  magnetizing  and  energy  components.  These  currents 
in  equivalent  single-phase  values  are  laid  off  in  Fig.  120  at  PD 
and  NC,  the  line  ZN  being  constructed  parallel  to  the  X  axis', 
draw  the  line  ZC.  This  will  be  a  chord  of  the  semicircle  and  the 
intersection  of  its  perpendicular  bisector  with  the  line  QN  will 
give  the  center  of  the 
semicircle  at  0' .  Then 
the  semicircle  is  con- 
structed with  a  radius  = 
OZ.  The  length  of  the 
line  CN  multiplied  by 
rated  voltage  of  motor 
gives  the  input  with  nor- 
mal voltage  and  locked 
motor  and  so  is  all  PR 
loss.  The  division  of  this 
loss  between  stator  and 

rotor  may  be  found  by  measuring  the  ohmic  resistance  of  the  rotor 
(equivalent  single  phase)  and  subtracting  this  value  from  the  total 
resistance  calculated  from  the  locked-saturation  curve  results. 

.   ,  „. .  , .    CH      rotor  resistance 

Then  the  point  H  is  so  taken  that  the  ratio  77^7  =  -    -p — r—     — 

CN      total  resistance 

There  are  several  assumptions  made  in  constructing  this 
diagram  which  make  it  inaccurate  for  the  prediction  of  points 
past  maximum  output.  As  the  diagram  is  only  used  over 
perhaps  60°  of  arc  these  discrepancies  do  not  show  themselves 
appreciably  and  this  method  of  determining  the  characteristics 
of  the  motor  is  probably  more  accurate  than  by  loading. 

Note.  —  It  is  to  be  noticed  that  there  is,  in  this  diagram,  a  peculiar  com- 
bination of  vectors.  From  the  nature  of  a  vector  representing  an  alternating 
current  it  is  evident  that  two  vectors  can  be  combined  to  give  a  resultant 
vector  only  when  the  two  have  the  same  frequency.  Yet  in  this  diagram  the 
rotor  current  ZE  has  a  frequency  of  s/  and  the  stator  current  has  a  frequency 
of/.  How  then  can  they  be  combined  as  vectors? 


Current  o 

V 

~ 

H 

x 

x. 

f' 

X 

/' 

X 

^X" 

J^ 

_. 



—f 

^-^-Ra 

b 

.1 

( 

11 

]' 

a 

T 

^ 

X 

x 

^x 

X 

)                                                                A 
Impressed  Voltage 

FIG.  121. 

196  ALTERNATING  CURRENTS 

At  any  value  of  the  stator  current  /,  the  rotor  current  is  given 
by  /i,  and  the  running  light  current  by  70.  On  the  diagram, 
currents  measured  along  the  X  axis  are  wattless  and  those 
measured  along  the  Y  axis  are  watt  components.  (The  diagram 
is  constructed  for  a  motor  having  the  same  number  of  turns  per 
circuit  in  the  rotor  and  stator.  If  such  is  not  the  case  the  vector 
addition  cannot  be  directly  used.) 

The  power  factor  of  the  motor  is  given  by  the  ratio  of  its  watt 
current  to  total  current,  or  is  cos  0  for  the  value  of  primary 
current  =  /.  The  rotor  power  factor  =  cos  <f>* '.  The  power 
factor  of  the  motor  is  most  readily  obtained  by  constructing  a 
quadrant  of  a  circle  with  center  at  0.  Have  the  radius  OK  = 
100  units  of  the  section  paper  on  which  the  diagram  is  con- 
structed. Then  the  power  factor  is  directly  obtained  by  pro- 
jecting to  the  Y  axis  the  point  of  intersection  of  the  primary 
current  with  the  circular  arc.  In  Fig.  120,  e.g.,  the  power  factor 
of  the  motor  when  taking  current  =  /  is  obtained  by  projecting 
the  point  E  to  the  Y  axis,  as  at  /,  and  the  value  of  cos  <j>  read 
directly  on  the  scale  of  which  OK  =  100. 

For  the  same  value  of  primary  current  the  slip  in  per  cent  is 

FT1 
obtained  by  the  ratio  of  -=j~  and  motor  efficiency  is  obtained  by 

.c/Cr 
EF 

the  ratio  of  ^~  •     The  rotor  PR  loss  is  obtained  by  the  length  of 

FG  and  the  stator  copper  loss  by  the  length  GM.  The  energy 
current  to  supply  stray-power  losses  is  given  by  OQ  and  the 
magnetizing  current  by  OP  and  the  leakage  current  by  PB. 
Torque  in  pound-feet  is  given  by  thfc  expression 

EF  X  7.05  EF  X  7.05 


syn.  speed  (1  -  s)  ,  L       FG\ 

syn.  speed  ^1  -  ^  j 


Maximum  torque  obtainable  is  found  at  the  point  W  and  is 
obtained  by  using  above  formula  for  current  OW.  W  is  ob- 
tained by  constructing  a  radius  of  the  semicircle  perpendicular 
to  the  line  ZH  and  W'}  the  point  of  operation  for  maximum 
output,  is  obtained  by  constructing  a  radius  perpendicular  to 
the  line  ZC. 

Perform  Experiment  34  on  the  same  motor  as  was  used  for 

*  The  student  may  derive  this  expression  easily  or  look  it  up  in  some  such 
book  as  McAllister's  "  A.C.  Motors." 


THE  POLYPHASE  INDUCTION  MOTOR  .      197 

Experiment  33  if  possible.  Obtain  necessary  data  to  construct 
circle  diagram.  Construct  this  diagram  carefully  upon  section 
paper  and  obtain  from  this  diagram  the  following  curves:  power 
factor,  efficiency,  speed,  torque,  and  plot  them  against  H.P. 
output  as  abscissae.  Obtain  points  at  about  every  J  rated  load 
up  to  150  per  cent  rated  load.  Predict  maximum  torque  and 
maximum  output  of  the  motor. 

Note.  —  The  circle  diagram  must  be  very  carefully  constructed  and  should 
have  a  diameter  at  least  8  inches. 


EXPERIMENT   XXXV. 

THE  VARIABLE-SPEED  INDUCTION  MOTOR,  ITS  CHARACTERISTICS 

BY  TEST  AND  CIRCLE  DIAGRAM;  VARIATION  OF  STARTING 

TORQUE  WITH  ROTOR  RESISTANCE. 

WHILE  an  induction  motor  is  essentially  a  constant-speed 
machine,  still,  for  special  purposes,  it  may  be  used  as  a  variable- 
speed  motor  by  introducing  resistance  in  the  rotor  circuit.  Its 
use  under  such  conditions  is  very  inefficient,  the  method  being 
exactly  analogous  to  the  speed  control  of  a  shunt  D.C.  motor  by 
introduction  of  resistance  in  its  armature  circuit.  In  using  the 
induction  motor  as  a  variable-speed  device,  resistance  is  intro- 
duced into  the  rotor  circuit  and  hence  the  rotors  of  such  motors 
must  be  wound  and  attached  to  slip  rings  from  which  the  rotor  cir- 
cuit is  closed  through  the  rotor  rheostat  and  its  controlling  switch. 

The  speed  control  of  an  induction  motor  by  this  method 
results  in  both  decreased  efficiency  and  maximum  output.  It  is 
seen  at  once  that  whatever  heat  is  dissipated  in  the  rotor  rheostat 
is  so  much  waste  energy,  and  as  the  motor  has  the  same  maximum 
torque  whatever  the  resistance  of  its  rotor  circuit,  any  decrease 
in  speed  for  a  given  torque  must  result  in  corresponding  decrease 
in  maximum  output.  The  construction  of  the  circle  diagram  for 
a  motor  having  various  resistances  in  its  rotor  circuit  is  evidently 
the  same  as  would  be  employed  if  the  rotor  $yas  short-circuited. 


Tjl 

The  diameter   of  the  semicircle  still   equals  —    —  ,  and  this 

Xi  +  X*' 

value  is  independent  of  r2.  In  Fig.  122  the  diagram  of  Fig.  120 
is  shown  and  this  diagram  will  hold  good  for  the  same  motor 
even  when  the  resistance  is  added  in  its  rotor  circuit.  The 
running  light  current  will  still  be  OZ,  the  same  value  it  had  for 
short-circuit  rotor;  but  the  point  C  will  be  changed.  If  resist- 
ance is  added  to  the  rotor  circuit  then  the  current  value  ascer- 
tained for  the  locked  saturation  curve  will  be  less  than  before 
and  the  extrapolated  value  of  locked  rotor  current  under  normal 
impressed  voltage  will  be  now  only  OC",  smaller  than  it  was 
before  but  still  on  the  same  semicircle. 

198 


THE  POLYPHASE  INDUCTION  MOTOR 


199 


For  further  increase  of  resistance  in  the  rotor  circuit,  the  point 
C  moves  to  C",  giving  a  still  smaller  locked  rotor  current.  .  The 
maximum  torque  is  fixed  by  the  line  WE  (O'W  being  perpendicu- 
lar to  ZB)  and  it  is  evident  that  the  rotor  resistance  does  not 
affect  the  maximum  obtainable  torque  until  the  external  resist- 
ance added  is  sufficient  to  bring  the  point  C  to  W.  With  this 
value  of  rotor  resistance,  maximum  torque  is  obtained  at  stand- 
still; any  further  increase  in  resistance  would  require  a  back- 
ward rotation  to  develop  maximum  torque. 

For  a  given  primary  current  the  torque  is  evidently  the  same, 
i.e.,  independent  of  rotor  resistance.  Also  the  power  factor  for 
a  given  primary  current  is  independent  of  rotor  resistance.  For 


D" 
FIG.  122. 


D' 


D 


a  given  input  (i.e.,  for  a  given  torque)  the  slip  is  directly  pro- 
portional to  the  total  rotor  resistance,  while  the  efficiency  and 
maximum  output  decrease  with  increasing  resistance.  This  is 
exactly  analogous  to  the  action  of  a  resistance  put  hi  series  with 
the  armature  of  a  D.C.  shunt  motor  for  speed  control. 

The  effects  of  added  rotor  resistance  upon  the  running  char- 
acteristics of  the  motor  are,  decrease  of  speed  directly  with  the 
amount  of  added  resistance,  decrease  of  efficiency  for  a  given 
output  and  decrease  in  a  maximum  possible  output.  The 
effect  upon  the  starting  characteristics  of  the  motor  are,  increased 
torque  (until  r2  reaches  a  value  equal  to  x2,  after  which  torque 
again  decreases),  increased  power  factor,  and  decrease  in  start- 
ing current. 

In  Fig.  123  are  given  two  sets  of  curves  to  show  the  effect  of 
resistance  upon  the  running  characteristics  of  a  motor,  the  full- 
line  curves  being  for  a  short-circuited  rotor  and  the  dotted 
curves  for  rotor  with  added  external  resistance. 


200 


ALTERNATING  CURRENTS 


In  Fig.  124  are  shown  the  effects  of  added  rotor  resistance 
upon  the  starting  characteristics  of  an  induction  motor. 

There  are  other  methods  for  controlling  the  speed  of  induction 
motors.  When  used  for  railway  purposes  they  may  be  used  in  a 


A ,  A=  Speed 

5,  B—  Effeciency 

C,  C'-  Power  factor 


Maximum  output,  with 
added  resistance 

I    I    I    I   I    I    I    I    II   I. 


Output 
FIG.  123. 

'kind  of  series  connection,  called  concatenation,  or  cascade  con- 
trol. In  this  method  the  rotor  current  of  one  motor  supplies  the 
power  for  the  stator  of  the  second,  the  rotor  of  the  second  being 
short-circuited  or  may  be  used  with  external  resistance.  The 
stator  of  the  first  motor  is  connected  to  the  line. 


Xorq-i 


Rotor  circuit  resistance 
FIG.  124. 

The  motor  may  have  its  stator  wound  in  such  a  manner  that 
the  number  of  poles  is  varied  by  proper  switching  arrangements. 
Both  of  these  methods  are  more  efficient  than  the  control  by 
rotor  resistance. 

The  frequency  of  the  power  supply  to  the  induction  motor  may 
be  varied,  although  this  is  not  a  practical  scheme  commercially, 
because  usually  power  is  distributed  at  only  one  frequency. 


THE  POLYPHASE  INDUCTION  MOTOR  201 

The  voltage  supplied  to  the  motor  may  be  varied  either  by  a 
compensator  or  an  adjustable  resistance.  Both  of  these  schemes 
give  very  poor  regulation  and  are  used  only  hi  exceptional  cases. 

Using  a  prony  brake  or  generator  for  load,  take  a  series  of  runs 
(from  no-load  to  1J  full-load  current)  on  an  induction  motor, 
using  different  resistances  in  the  rotor  circuit  for  each  run. 
Read  volts,  amperes  and  watts  input,  speed,  frequency  and 
torque.  Have  voltage  and  frequency  at  rated  values.  Take 
one  run  with  the  rotor  short-circuited.  Take  proper  measure- 
ments for  the  construction  of  the  circle  diagram  of  the  motor 
under  the  different  conditions. 

With  about  \  rated  voltage  impressed  hi  the  stator  and  the 
rotor  locked  take  a  series  of  readings  to  show  the  effect  upon 
the  starting  characteristics  of  the  motor  as  the  rotor  resistance 
is  varied  from  its  maximum  value  to  short-circuited  rotor. 

Read  volts,  amperes,  watts  and  torque. 

Construct  a  circle  diagram  for  the  prediction  of  the  running 
characteristics  of  the  motor  with  its  different  rotor  resistance. 

Plot  curves  (from  test  results)  of  efficiency,  power  factor, 
speed,  torque  and  primary  current  for  the  different  resistances. 
All  curves  to  be  plotted  with  H.P.  output  as  abscissa.  Upon 
these  curves  indicate  (using  different  color  of  ink)  the  points  as 
determined  from  the  circle  diagram. 

Upon  another  sheet  plot  curves  of  starting  torque,  starting 
current  and  power  factor,  using  total  rotor-circuit  resistance  as 
abscissa. 

All  of  above  curves  are  to  be  plotted  with  equivalent  single- 
phase  values  of  the  quantities  involved. 

If  time  permits  try  one  of  the  other  methods  mentioned  for 
speed  control  of  induction  motors. 


EXPERIMENT   XXXVI. 

THE  SINGLE-PHASE  INDUCTION  MOTOR;  ITS  CHARACTERISTICS; 

APPLICATION  OF  CIRCLE  DIAGRAM  FOR  PREDETERMINATION; 

STARTING  AS  A  REPULSION  MOTOR. 

THE  question  of  the  rotating  magnetic  field  is  not  quite  so 
readily  solved  for  the  single-phase  induction  motor  as  it  is  for 
the  polyphase  motor.  It  is  not  directly  evident  how  one  set  of 
magnetizing  coils,  supplied  with  single-phase  current,  can  pro- 
duce a  magnetic  field  of  practically  uniform  strength,  rotating 
in  space.  As  a  matter  of  fact,  the  stator  coils  of  such  a  motor 
can  produce  only  an  alternating  field,  stationary  in  space,  and 
it  is  only  by  the  reactions  of  the  currents  in  the  rotating  second- 
ary circuits  that  a  rotating  magnetic  field  is  produced.* 

After  the  rotating  field  has  been  proved  then  the  treatment 
of  the  single-phase  induction  motor  is  essentially  the  same  as 
that  of  the  polyphase  motor. 

The  principle  characteristic  of  the  single-phase  motor  is  the 
absence  of  starting  torque.  The  reactions  of  the  rotor  currents 
to  produce  a  rotating  magnetic  field  only  occur  after  the  rotor 
has  acquired  some  angular  velocity.  Such  a  motor  will  not 
start  by  itself  but  if  the  rotor  is  given  a  start  (and  there  is  no 
load  on  the  motor)  then  the  motor  will  accelerate  and  come  up 
to  nearly  synchronous  speed  in  the  same  direction  as  the  rotor  is 
started.  The  operation  of  the  motor  is  just  the  same  in  whatever 
direction  it  revolves,  and  this  is  determined  by  the  starting  effort. 

Compared  with  the  polyphase  motor  it  may  be  said  in  general 
that  the  power  factor,  efficiency  and  pull-out  point  are  all  some- 
what lower  for  the  single-phase  machine,  while  the  speed  regula- 
tion is  somewhat  better. 

If  the  stator  coils  of  a  polyphase  motor  produce  a  rotating 
field  of  constant  strength,  then  at  synchronous  speed  there  will 
be  no  currents  in  the  rotor  coils.  With  the  single-phase  motor 
there  is  a  large  magnetizing  current  of  double  line  frequency 
flowing  in  the  rotor  at  synchronous  speed;  owing  to  this  fact 
there  is  considerable  rotor  PR  loss  at  all  speeds,  while  in  a  poly- 

*  See  Crocker  and  Arendt,  "  Electric  Motors,"  page  213,  et  seq. 

202 


THE  SINGLE-PHASE  INDUCTION  MOTOR 


203 


phase  motor  this  loss  is  negligible  at  synchronous  speed.  At  any 
speed  other  than  synchronous  there  will  be  in  the  rotor  a  combina- 
tion of  this  magnetizing  current  and  an  energy  component  depend- 
ing upon  the  load.  Hence  for  given  load  the  rotor  PR  losses  are 
greater  in  the  single-phase  than  in  the  polyphase  motor.  For  a 
given  capacity  a  single-phase  motor  must  have  more  iron  and 
copper  than  a  polyphase  motor,  resulting  in  slightly  greater  stator 
core  and  PR  losses.  These  three  effects  result  in  giving  the 
polyphase  motor  a  slightly  greater  efficiency  than  the  single  phase. 

The  decreased  "  pull-out  "  point  of  the  single-phase  motor  is 
explained  by  the  fact  that  as  the  motor  slows  down  the  quadra- 
ture component  of  its  field  decreases  and  so  results  in  decreased 
average  field  strength  and  so  a  decreased  torque. 

As  many  single-phase  induction  motors  are  started  as  repul- 
sion motors  the  underlying  principle  of  this  type  of  motor  will  be 
briefly  discussed.  The  arma- 
ture of  such  a  motor  is  drum 
wound  and  connected  to  a  com- 
mutator, exactly  similar  to  an 
ordinary  D.C.  motor  arma- 
ture; in  the  repulsion  motor, 
however,  the  brushes  are  con- 
nected directly  together  and 
the  armature  is  not  electrically 
connected  to  the  power  supply. 
Now  the  E.M.F.  and  torque 
generated  by  such  an  arma- 
ture can  most  easily  be  dis- 
cussed  by  supposing  the  arma- 

ture winding  to  consist  of  only  one  turn,  this  turn  being  constantly 
at  right  angles  to  the  commutating  plane.  Such  an  armature  is 
sketched  in  Fig.  125.  Now,  if  the  brushes  are  so  placed  on  the 
commutator  that  the  armature  winding  is  represented  by  the  coil 
(a)  it  is  evident  that  there  will  be  no  current  set  up  in  the  coil 
due  to  the  alternation  of  the  magnetic  field  because  the  flux  does 
not  cut  the  coil,  the  flux  being  parallel  to  the  plane  of  the  coil. 

If,  however,  the  brushes  on  the  commutator  are  shifted  90° 
(electrical),  so  that  the  armature  winding  may  be  represented 
by  the  coil  (6)  a  heavy  current  will  flow  because  the  winding  acts 
like  the  short-circuited  secondary  of  a  transformer.  In  this 
position,  however,  armature  can  exert  no  torque  because  the  sides 


pIG 


204  ALTERNATING  CURRENTS 

of  the  coil  6  can  move  only  in  a  direction  parallel  to  the  field. 
Hence  with  the  brushes  in  either  position  a  or  b  the  rotor  will  exert 
no  turning  effort.  Now,  if  the  brushes  are  set  at  some  interme- 
diate position  (so  that  the  armature  winding  is  represented  by 
coil  c)  a  current  will  flow  in  the  coil  and  also  the  coil  sides  are  in 
a  field  so  that  torque  will  be  exerted,  tending  to  turn  the  arma- 
ture in,  say,  clockwise  direction.  At  the  next  alternation  of  the 
field  the  current  in  the  armature  will  reverse  but  the  field  will 
also  be  reversed  and  so  the  torque  will  again  be  in  clockwise  direc- 
tion. The  result  is  a  pulsating  unidirectional  torque  and  such  an 
A.C.  motor  gives  characteristics  very  similar  to  those  of  the  D.C. 
series  motor,  giving  a  heavy  starting  torque,  decreasing  its  torque 
and  current  as  the  speed  increases  and  having  no  fixed  speed  limit. 

The  magnitudes  of  the  starting  torque  and  current  depend 
upon  how  much  the  brushes  are  shifted  from  their  neutral 
position,  i.e.,  the  angle  6  in  Fig.  125. 

The  Wagner  single-phase  induction  motor  is  equipped  with 
commutator  and  short-circuited  brushes  and  is  started  as  a 
repulsion  motor,  giving  a  starting  torque  considerably  greater 
than  full-load  running  torque  if  so  desired.  When  the  motor 
approaches  synchronous  speed,  a  centrifugal  device  throws  off  the 
brushes  and  short-circuits  all  of  the  commutator  bars,  thereby 
changing  the  armature  to  a  squirrel-cage  rotor.  The  machine 
then  operates  as  a  single-phase  induction  motor. 

If  the  brushes  and  centrifugal  governor  are  correctly  set,  the 
transition  from  repulsion  motor  to  induction  motor  takes  place 
without  too  violent  a  change  in  either  torque  or  current.  In 
Fig.  126  in  dotted  lines  are  given  the  speed-torque  and  speed- 
current  curves  of  the  motor  when  operating  on  the  repulsion 
principle  and  the  full  lines  give  the  same  curves  for  the  motor 
operating  as  a  single-phase  induction  motor.  The  proper  speed 
for  the  centrifugal  switch  to  act  is  indicated  in  Fig.  126;  this  speed, 
it  will  be  noticed,  is  such  that  the  repulsion  motor  still  has  some- 
what greater  than  full-load  torque  when  the  switch  changes  the 
motor  to  the  induction  principle.  The  repulsion  motor  character- 
istics may  be  readily  altered  by  changing  the  angle  at  which  the 
brushes  are  set  on  the  commutator. 

Study  the  construction  and  operation  of  a  Wagner  single- 
phase  motor.  Take  a  load  run  to  get  power  factor,  speed  and 
efficiency  as  related  to  the  output  of  the  motor. 

Try  the  effect  on  starting  torque  and  current  of  changing  the 


THE  SINGLE-PHASE  INDUCTION  MOTOR 


205 


angular  position  of  the  brushes  (this  test  may  be  run  at  50  per 
cent  rated  voltage  so  that  motor  will  not  overheat) .  With  about 
50  per  cent  rated  voltage  on  the  motor  terminals  obtain  its 
characteristics  as  a  repulsion  motor  from  standstill  to  that  speed 
at  which  the  brushes  throw  off. 

Obtain  sufficient  data  to  construct  the  circle  diagram  for  this 
motor.  The  centrifugal  device  must  be  clamped  while  taking 
the  locked-saturation  curve.  From  the  circle  diagram  get  the 
curves  given  in  Fig.  126  and  see  how  nearly  the  conditions  be- 
tween repulsion  and  induction  characteristics  of  the  motor  tested 


30         40         50         60         70         80 
Per  cent  synchronous  speed 

FIG.  126. 


90       100 


110 


agree  with  the  curves  given  in  Fig.  126.  If  they  do  not  agree 
what  would  you  do  to  the  motor  to  make  it  act  as  described  in 
the  discussion  of  Fig.  126? 

On  one  sheet  of  section  paper  plot  curves  of  efficiency,  power 
factor,  speed  and  current  against  H.P.  output  as  abscissa,  using 
test  results.  On  another  sheet  construct  the  circle  diagram  and 
plot  the  values  of  primary  current  obtained  from  test  to  see  how 
nearly  they  lie  in  the  circular  locus.  To  lay  off  these  values 
lay  off  the  energy  component  of  current  up  the  Y  axis  of  the 
circle  diagram,  and  through  this  point  draw  a  line  parallel  to  the 
X  axis.  Then  with  0  as  center  and  primary  current  as  radius, 
intersect  this  line.  Intersection  is  point  desired. 

Upon  another  cross-section  sheet  plot  curves  similar  to  those 
of  Fig.  126  and  mark  speed  at  which  brushes  throw  off. 


EXPERIMENT  XXXVII. 

STUDY  OF  THE  INDUCTION  GENERATOR;  MAGNETIZATION 

CURVE;    EXTERNAL   CHARACTERISTIC  WHEN  EXCITED 

BY  SYNCHRONOUS  MOTOR;    CHANGE  FROM 

MOTOR  ACTION  TO  GENERATOR  ACTION 

WITH  VARIATION  OF  SPEED,  WHEN 

CONNECTED  TO  SUPPLY  OF 

CONSTANT  FREQUENCY. 

IF  the  rotating  field  produced  by  the  stator  winding  of  a  poly- 
phase induction  motor  is  such  that  its  density  may  be  represented 
by  0m  cos  a,  where  <£m  is  the  maximum  density  and  a  is  the  angle 
between  the  point  considered  and  the  axis  of  the  field,  then  when 
the  rotor  is  turning  at  synchronous  speed  there  will  be  no  current 
flowing  in  the  rotor  conductors,  as  each  conductor  will  continu- 
ally occupy  a  position  where  the  magnetic  field  has  a  constant 
value  and  so  the  conductor  generates  no  E.M.F.  If  the  stator 
coils  are  not  so  distributed  that  they  produce  a  field  of  cosine  dis- 
tribution, this  statement  is  not  true;  even  at  synchronous  speed 
currents  will  flow  in  the  rotor  circuits  which,  by  their  reaction 
upon  the  stator  field,  transform  it  into  a  field  of  cosine  distribution. 

With  the  rotor  turning  at  synchronous  speed,  the  current  in 
the  stator  will  have  a  magnetizing  component  of  sufficient 
strength  to  produce  a  field  which  by  its  change  gives  a  C. E.M.F. 
practically  equal  to  the  impressed  E.M.F.,  and  an  energy  com- 
ponent sufficient  to  furnish  the  iron  losses  of  the  motor. 

Now,  no  matter  what  the  speed  of  rotation  may  be  the  current 
relations  in  the  stator  and  rotor  must  be  the  same  as  exist 
between  the  primary  and  secondary  current  of  a  static  trans- 
former. When  an  energy  or  wattless  current  is  flowing  in  the 
secondary  the  primary  must  also  be  carrying  an  equal  current 
plus  its  no-load  current.  Hence,  when  the  rotor  of  an  induction 
motor  is  carrying  a  certain  current  the  stator  must  be  carrying 
the  same  current  (or  rather  an  equal  current  in  the  opposite 
direction)  plus  its  no-load  current  (1  : 1  windings  considered) . 

If  the  rotor  of  an  induction  motor  is  turning  at  less  speed  than 
the  magnetic  field  it  is  said  to  have  a  positive  slip.  The  rotor 

206 


THE  INDUCTION  GENERATOR          207 

conductors,  cutting  flux,  will  generate  what  we  will  call  a  positive 
E.M.F.  and  there  will  flow  in  the  rotor  circuits  a  current  which 
will  be  nearly  in  phase  with  the  rotor  E.M.F.  The  lag  of  the 

current  will  be  tan"1  =  — -  and  for  small  values  of  s  this  angle 

7*2 

will  be  small.  This  rotor  current  must  have  its  counterpart 
in  the-stator  and  if  the  slip  is  positive  this  stator  current  will 
be  such  that  its  product  with  the  impressed  E.M.F.  repre- 
sents power  input  to  the  machine,  and  it  is  absorbing  electric 
power  and  giving  out  mechanical  power,  i.e.,  is  running  as  a 
motor. 

If  the  rotor  is  connected  to  some  driving  device  (e.g.,  another 
motor)  and  is  driven  at  some  speed  higher  than  synchronous,  it  is 
at  once  evident  that  the  E.M.F.  generated  in  the  rotor  conduc- 
tors will  be  in  the  opposite  direction  to  what  it  previously  had. 
Hence  the  direction  of  current  in  the  rotor  will  be  opposite  and 
so  must  its  counterpart  in  the  stator  windings.  This  means, 
of  course,  that  the  current  in  the  stator,  due  to  the  rotor  currents, 
is  in  a  direction  opposite  to  what  it  had  when  the  machine  was 
operating  as  a  motor;  the  energy  component  of  stator  current 
is,  for  this  condition,  flowing  in  a  direction  opposite  to  the  im- 
pressed E.M.F.  Hence,  the  machine  must  be  giving  off  power 
and  so  is  a  generator.  Such  a  machine  is  called  an  induction 
generator;  as  the  frequency  of  current  delivered  is  not  equal  to 
the  rotational  frequency  of  the  machine,  it  is  sometimes  called 
the  asynchronous  generator. 

As  the  slip  of  an  induction  machine  changes  from  positive  to 
negative,  it  has  just  been  shown  that  the  energy  component  of 
the  stator  current  changes  phase  180°  with  respect  to  the  im- 
pressed E.M.F. 

The  wattless  or  magnetizing  component  of  the  stator  remains 
in  the  same  phase,  however.  With  respect  to  the  impressed 
E.M.F.  of  the  motor,  this  is  a  lagging  current;  reckoned  from  the 
phase  of  the  motor  C. E.M.F.  it  would,  therefore,  be  a  leading 
current,  so  that  referred  to  the  phase  of  the  generated  E.M.F.  of 
the  asynchronous  generator  its  magnetizing  current  is  a  leading 
current.  For  the  induction  generator  to  be  operative,  therefore, 
it  must  be  connected  to  a  line  of  certain  frequency  and  some 
device  in  the  line  must  be  capable  of  drawing  a  leading  current 
from  the  line,  the  magnitude  of  this  leading  current  being  the 
ainount  of  magnetizing  current  required  by  the  generator. 


208 


ALTERNATING  CURRENTS 


The  magnitude  of  rotor  current,  and  hence  of  the  stator  cur- 
rent, depends  upon  the  amount  of  negative  slip.  The  voltage 
of  the  generator  depends  upon  the  strength  of  its  field,  i.e.,  upon 
the  amount  of  magnetizing  current  furnished  to  it.  These  two 
characteristics  are  the  distinguishing  features  of  the  induction 
generator. 

An  over-excited  synchronous  motor  has  both  of  the  features 
necessary  to  make  an  induction  generator  operate;  it  has  a 
definite  frequency  and  draws  a  leading  current.  If,  therefore,  a 
synchronous  motor,  with  sufficient  field  excitation,  is  connected 
to  the  terminals  of  an  induction  machine  of  higher  rotational 
frequency  than  the  motor,  the  induction  machine  will  generate, 
supplying  the  amount  of  power  to  run  the  motor,  and  other  load 
(as  e.g.,  lamps)  may  be  operated  from  the  line. 

In  starting,  the  two  machines  may  be  electrically  connected 
and  both  be  brought  up  to  a  speed  somewhere  near  the  proper 
running  speed  of  the  induction  machine,  then  the  driving  power 
may  be  taken  from  the  synchronous  machine  and  it  will,  if 
sufficiently  excited,  continue  to  run,  drawing  sufficient  power 
from  the  generator  to  supply  its  stray-power  losses  and  taking 
from  the  line  a  leading  current  which  becomes  the  magnetizing 
current  for  the  generator. 

As  may  be  seen  from  this  discussion  the  current  relations  in 
the  induction  generator  follow  the  same  laws  as  in  the  motor 

and  so  the  circle  diagram 
must  hold  good  for  both  if 
it  holds  for  the  motor.  In 
Fig.  127  the  application  of 
the  circle  diagram  to  the 
generator  is  indicated.  The 
no-load  current  is  shown  at 
OP;  for  motor  action  some 
current  as  OA  is  required, 
having  an  energy  component 
OC  and  magnetizing  compo- 
nent OE.  The  current  OD 
is  required  to  furnish  the  no- 

FlG  12y  load  losses.     If  not   loaded 

and  not    speeded  up  by  a 

driver  the  machine  operates  at  perhaps  99.5  per  cent  synchronism 
and  draws  an  energy  current  OD  from  the  line. 


THE  INDUCTION  GENERATOR  209 

As  the  rotor  is  speeded  up  part  of  the  energy  required  to 
supply  machine  losses  is  supplied  by  the  driver  mechanically, 
and  if  the  speed  is  increased  to  perhaps  100.5  per  cent  syn- 
chronism, the  stator  current  becomes  OH,  purely  a  magnetizing 
current,  and  all  of  the  machine  losses  are  supplied  mechanically. 
From  99.5  per  cent  to  100.5  *  per  cent  synchronism  the  machine 
gives  neither  mechanical  nor  electrical  power;  this  is  called  the 
region  of  total  loss.  At  some  speed  above  100.5  per  cent  syn- 
chronism the  energy  component  of  stator  current  may  go  to  some 
value  as  OE,  in  phase,  opposite  to  OC.  The  total  stator  current 
OB  will  be  found  on  the  same  circular  locus  as  OA.  The  com- 
plete application  of  the  circle  diagram  to  induction  generator 
characteristics  will  be  given  later. 

An  understanding  of  some  of  the  generator  characteristics  may 
be  obtained  by  considering  what  happens  when  a  generator, 
excited  by  a  floating  synchronous  motor,  is  loaded.  Suppose 
that  the  speed  of  the  generator  is  adjusted  to  give  a  speed  of 
60  cycles  to  the  synchronous  motor  and  the  synchronous  moto- 
field  is  adjusted  to  give  the  induction  generator  a  terminal  volt- 
age of  110  volts.  If  the  generator  rotor  is  held  at  constant  speed 
and  some  load  is  connected  to  the  generator  terminals  two  effects 
will  be  observed.  Both  the  voltage  and  frequency  of  the  system 
will  decrease.  Even  if  the  generator  induced  voltage  remained 
constant  the  terminal  voltage  would  fall  with  increase  of  load 
due  to  the  increased  IZ  drop  in  the  stator  windings.  But  the 
induced  voltage  itself  decreases  with  increase  of  load.  The  lag 

|Mg 

angle  of  the  rotor  current  is  tan"1  =  -  -  and,  due  to  the  slowing 

down  of  the  synchronous  motor  (explained  below),  s  increases 
with  load  so  that  the  rotor  currents  tend  to  demagnetize  the 
stator  more  and  more  as  the  load  is  increased.  This  may  be 
readily  seen  from  Fig.  127.  To  give  110  volts  induced  voltage, 
e.g.,  at  synchronous  speed  requires  a  magnetizing  current  of  OH. 
When  the  load  on  the  generator  is  equal  to  OE,  then  the  mag- 
netizing current  necessary  to  give  110  volts  induced  E.M.F.  is 
given  by  the  vector  OR.  This  current  is  made  up  of  two  parts; 
the  original  current  OH  and  the  demagnetizing  component 
of  the  rotor  current  HR.  Hence,  to  maintain  constant  induced 
voltage  as  the  load  increases  the  synchronous  motor  must  be 

*  Of  course  these  numerical  values  of  slip  are  only  approximate;  they  will 
be  different  in  different  machines. 


210  ALTERNATING  CURRENTS 

so  excited  that  at  any  generator  load  the  motor  draws  a  lead- 
ing current  just  equal  to  this  wattless  component  of  the  stator 
current. 

The  synchronous  motor  slows  down  because  it  requires  a  cer- 
tain amount  of  power  to  run  itself,  and  if  the  terminal  voltage  of 
the  generator  falls  the  C.E.M.F.  of  the  synchronous  motor  must 
also  decrease  to  permit  the  required  current  to  flow.  The  slowing 
down  of  the  synchronous  motor  may  be  explained  from  another 
standpoint.  The  stator  current  (load  current)  can  only  increase 
if  the  rotor  current  increases,  as  before  noted.  The  rotor  current 
can  only  increase  if  its  slip  increases  and  as  the  rotor  is  being 
driven  at  constant  speed  its  slip  can  only  increase  by  a  slowing 
down  of  the  field  speed.  Hence  the  synchronous  motor  must 
slow  down  sufficiently  to  make  the  difference  between  field  speed 
and  rotor  speed  such  that  the  rotor  currents  generated  produce 
in  the  stator  just  that  current  demanded  by  the  load. 

It  will  be  noticed  that  this  behavior  of  an  induction  generator 
excited  by  a  floating  synchronous  motor  is  exactly  similar  to  that 
of  a  shunt-wound  D.C.  generator  with  its  brushes  set  in  such  a 
position  that  armature  reaction  demagnetizes  the  field.  As  load 
in  such  a  machine  increases  the  terminal  E.M.F.  drops  because 
of  the  increased  IR  drop  in  the  armature  windings  and  because 
the  induced  voltage  falls,  due  to  the  weakening  of  the  field  by 
the  armature  reaction.  If  it  is  desired  to  maintain  constant  ter- 
minal voltage  on  such  a  machine  the  shunt  field  current  must  be 
increased  sufficiently  to  overcome  both  of  these  effects.  In  the 
shunt  D.C.  machine  this  is  accomplished  by  cutting  out  the  field 
rheostat  and  in  the  case  of  the  induction  generator  it  is  accom- 
plished by  increasing  the  field  current  of  the  synchronous  motor. 
The  decrease  in  frequency  as  the  load  increases  can  only  be 
overcome  by  speeding  up  the  rotor.  With  increase  of  load  there 
must  be  corresponding  increase  in  slip  and  if  the  rotor  turns  at 
constant  speed  the  synchronous  motor  (by  the  speed  of  which 
the  frequency  of  the  power  generated  is  determined)  must  slow 
down. 

The  magnetization  curve  of  the  asynchronous  generator  can 
readily  be  obtained  as  follows:  With  the  synchronous  motor 
floating  on  the  generator  line  and  no  load  on  line,  run  the  gene- 
rator at  such  speed  that  the  motor  constantly  gives  the  fre- 
quency at  which  the  magnetization  curve  of  the  generator  is 
desired.  By  means  of  an  ammeter,  wattmeter  and  voltmeter 


THE  INDUCTION  GENERATOR 


211 


in  the  line  connecting  the  two  machines  measure  the  variation 
in  terminal  E.M.F.  of  generator,  watts  and  current  as  the  field 
current  of  the  synchronous  motor  is  varied  through  that  range 
which  gives  an  E.M.F.  varying  between  rated  value  for  the 
generator  and  as  low  as  it  is  possible  to  go.  The  generator 
current  may  be  resolved  into  its  power  and  wattless  components 
and  the  relation  between  wattless,  or  magnetizing  current,  and 
terminal  E.M.F.  gives  approximately  the  magnetization  curve  of 
the  generator.  The  combination  becomes  nonoperative  at  low 
values  of  voltage  so  that  the  foot  of  the  magnetization  curve 
cannot  be  determined. 

An  interesting  feature  of  the  induction  generator  is  the  possi- 
bility of  excitation  by  condensers.  Although  of  not  much  com- 
mercial importance  at  present,  an  explanation  of  this  method  of 
operation  will  present 
the  characteristics  of 
the  machine  from  a  dif- 
ferent standpoint  and 
so  will  be  considered 
here. 

The  magnetization 
curve  of  the  generator 
will  have  the  general 
form  given  in  Fig.  128. 
The  impedance  of  a  con- 
denser is— ~  and  the  cur- 
rent in  such  a  condenser  9 

E 
is  —^-    The  current  is, 


Generated  E.M.F.  ^  ^  ^ 

^ 

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ly 

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7 

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—  — 

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— 

£*• 

-• 

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A 

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/ 

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/ 

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^ 

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/ 

/ 

/ 

/ 

/ 

/ 

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I 

'     / 

/ 

/ 

, 

/ 

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f 

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A~ 

/ 

// 

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y 

<i 

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j 

f 

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G 

Magnetizing  Current 

FIG.  128. 


therefore,  a  straight  line  function  of  the  voltage  and  so  may 
be  represented  by  a  straight  graph,  using  current  and  voltage 
for  the  abscissa  and  ordinate.  The  equation  of  the  graph  will 


be  I 


1 


E  tan  </>,  where  0  =  tan"1  -~ 


In  Fig.  128  three  such 

graphs  are  shown  at  OC,  OD  and  OE,  the  graph  OC  being  for  a 
condenser  of  greater  capacity  than  OD,  etc.  The  magnetization 
curve  of  the  generator  is  shown  by  the  curve  OAB.  If  the 
normal  voltage  of  the  generator  =  OB',  then  the  condenser 
whose  graph  is  OC  will  just  furnish  the  proper  amount  of  leading 
current,  OF,  to  generate  the  voltage  OB'.  A  smaller  condenser 


212 


ALTERNATING  CURRENTS 


whose  graph  is  OD  requires  a  voltage  of  OD'  to  draw  from  the 
line  a  leading  current  of  value  =  OF.  But  the  generator,  when 
furnished  with  magnetizing  current  OF,  can  generate  only 
voltage  =  OB',  which  will  cause  to  flow  in  the  condenser  a 
current  equal  to  OG.  Hence,  the  condenser  D  would  not  serve 
to  produce  in  the  generator  normal  voltage.  If  the  generator 
is  operating  with  condenser  C  at  normal  voltage  OB',  and  the 
condenser  D  is  substituted  for  C,  the  voltage  of  the  generator 
will  immediately  fall  to  OA',  at  which  voltage  the  magnetization 
curve  and  condenser  graph  intersect.  With  the  condenser  E 
the  generator  could  not  operate  because  at  no  voltage  does  the 
condenser  draw  enough  leading  current  to  magnetize  the  gene- 
rator to  that  same  voltage;  in  other  words,  the  magnetization 
curve  and  condenser  graph  do  not  cross.  When  once  brought 
up  to  voltage  either  of  the  condensers  C  and  D  will  serve  to 
maintain  the  generator  voltage,  but  it  is  quite  likely  that  if 
the  generator  is  running  idle  and  either  of  these  condensers 
should  be  connected  across  the  generator  terminals  it  would 
not  build  up. 

If  the  magnetization  curve  of  Fig.  128  is  much  magnified  and 
only  its  lower  extremity  considered  we  have  Fig.  129. 

It  is  seen  that  the  graphs  OC  and  OD  both  have  a  region  at 
the  lower  part  of  the  magnetization  curve  where  they  lie  above 

the  magnetization  curves  of  the 
generator.  Of  course,  the  genera- 
tor could  not  " build  up"  through 
these  regions  because  for  a  given 
magnetizing  current  the  condensers 
require  more  voltage  than  the  gen- 
erator can  give.  To  make  the 
generator  build  up  some  condenser, 
as  H,  must  be  used,  such  that  its 
graph  OH  does  not  intersect  the 
magnetization  curve  in  the  region 
shown  in  Fig.  129,  but  in  using  such 
a  condenser  the  generator  is  likely 
to  build  up  to  abnormal  values  of 
voltage,  so  as  soon  as  the  machine  begins  to  generate  the  con- 
denser must  be  cut  down  to  its  proper  value. 

When  using  condenser  for  excitation  it  is  not  at  once  evident 
how  the  frequency  of  the  generated  E.M.F.  is  determined. 


Magnetizing  Current 
FIG.  129. 


THE  INDUCTION  GENERATOR 


213 


The  generator  will  have  a  certain  equivalent  inductance  L* 
and  this,  combined  with  the  capacity  C,  makes  a  resonant 
circuit  whose  natural  period  of  oscillation  =  2  r  "X/LC  (neglect- 
ing the  effect  of  resistance).  Now,  it  may  be  ascertained  both 
mathematically  and  experimentally  that  the  condenser  used 
must  be  such  that  the  natural  period  of  the  system  is  less  than 
the  speed  of  rotation  of  the  generator.  This  only  emphasizes 
the  fundamental  principles  noted  before,  that  the  generator  slip 
must  be  negative.  The  speed  of  the  field  rotation  with  con- 
denser excitation  must  be  brought  lower  than  the  rotor  speed, 
and  this  is  done  by  so  increasing  C  that  the  natural  period  of  the 
system  has  some  value  lower  than  rotor  speed. 

The  commercial  application  of  induction  generators  is  not 
very  wide.  Some  polyphase  installations  for  railway  work 
utilize  the  generator  action  of  an  induction  motor  run  above 


B 


Motor 


Positive  Slip 


Negative  Slip 


'JO 


100 
Per  cent  Synchronous  Speed 

FIG.  130. 


synchronous  speed  for  a  braking  effect  when  on  down  grades. 
Of  course,  the  power  pumped  back  into  the  line  by  such  use  of 
the  motors  means  just  so  much  saved  energy,  as  the  load  on  the 
power  house  is  decreased.  Some  recent  installations  of  steam 
turbines,  utilizing  the  exhaust  of  reciprocating  engines,  have 
used  induction  generators.  Such  generators  are  connected 

*  A  more  complete  discussion  of  this  point  will  be  published  in  the  near 
future.  Prof.  Pupin  and  the  writer  have  done  a  lot  of  theoretical  and  experi- 
mental work  on  the  subject  and  expect  to  soon  publish  an  article  giving  the 
results  of  these  investigations. 


214 


ALTERNATING  CURRENTS 


directly  to  the  bus-bars,  the  necessary  leading  current  being 
obtained  by  overexciting  one  of  the  alternators.  The  induction 
generator  may  be  used  to  increase  the  output  capacity  of  a 
station  without  appreciably  increasing  the  short-circuit  risk  of 
the  station. 

As  the  power  factor  of  induction  machines  is  not  high,  the 
practicability  of  utilizing  condenser  excitation  is  not  marked. 
The  condensers  would  have  to  be  exceedingly  large. 

Figure  the  size  of  condenser  required  in  a  5000  K.V.A.,  60 
cycle  (E.M.F.  =  2300  volts)  station  of  induction  generators, 
supposing  the  power  factor  of  the  machines  run  as  motors  as 
93  per  cent  and  that  of  the  load  as  85  per  cent. 

The  complete  behavior  of  an  induction  generator  can  best  be 
obtained  from  such  a  set  of  curves  as  is  given  in  Fig.  130.  The 
continuous  change  in  the  different  quantities  from  motor  to 
generator  action  is  evident.  For  the  region  of  the  "  total  loss," 
previously  referred  to,  the  efficiency  is,  of  course,  zero.  By 
negative  power  factor  is  meant  that  the  magnetizing  current  is 


FIG.  131. 

leading  the  terminal  E.M.F.  of  the  generator.  In  this  set  of 
curves  the  letters  have  the  following  significance. 

A  =  torque,  B  =  efficiency,  C  =  total  stator  current,  D  =  power 
factor,  E  =  watt  component  of  stator  current,  F  =  wattless 
current. 

The  tests  to  be  made  in  this  experiment  are :  the  magnetization 
curve,  to  be  obtained  as  previously  described,  by  synchronous- 
motor  excitation;  external  characteristic,  using  synchronous- 
motor  excitation  and  constant  rotor  speed,  load  to  be  non- 
inductive;  operating  characteristics  given  in  Fig.  130;  sufficient 


THE  INDUCTION  GENERATOR          215 

data  to  construct  circle  locus  of  current  (this  data  is  obtained 
exactly  as  though  the  machine  being  tested  were  a  motor). 

To  obtain  the  operating  characteristics  as  given  in  Fig.  130, 
make  connections  as  in  Fig.  131.  The  D.C.  machine  should  be 
approximately  the  same  capacity  of  the  induction  machine,  and 
the  booster  in  the  D.C.  armature  circuit  capable  of  boosting  or 
crushing  the  D.C.  line  voltage  by  10  per  cent  or  15  per  cent. 
The  A.C.  generator  is  to  be  connected  to  an  A.C.  line  of  constant 
potential  and  frequency.  In  connecting  the  induction  generator 
to  the  A.C.  line  care  must  be  taken  that  the  rotating  field  goes 
in  the  same  direction  as  the  D.C.  motor  drives  the  rotor. 

Measure  the  core  loss  and  friction  of  the  driving  motor  at 
normal  field  current  at  such  speeds  as  give  to  the  induction 
generator,  90  per  cent,  100  per  cent  and  110  per  cent  of  rated 
speed.  Measure  D.C.  armature  resistance.  By  means  of  the 
booster  vary  the  action  of  the  A.C.  machine  over  as  wide  a  range 
as  possible,  through  both  motor  action  and  generator  action. 
Keep  the  field  current  of  the  driving  motor  constant  at  normal 
value  so  that  the  core  losses  may  be  obtained  from  core-loss  curve 
previously  measured.  The  PR  loss  in  the  D.C.  machine  can  be 
calculated  for  any  value  of  /. 

Read  D.C.  armature  current  and  volts,  and  A.C.  amperes, 
volts  and  watts.  Measure  speed  of  A.C.  machine  either  by  one 
of  the  slip  methods  previously  described  in  Experiment  33,  or,  if 
the  slip  becomes  too  large  for  this  method,  by  a  tachometer. 
For  each  load  calculate  the  input  or  output  of  the  induction 
machine  and  so  obtain  the  set  of  curves  called  for. 

Construct  circle  locus  of  current  and  plot  on  it  the  values  of 
current  measured  in  this  run.  Points  may  be  plotted  as  de- 
scribed in  Experiment  36. 


EXPERIMENT   XXXVIII. 
THE  SINGLE-PHASE  SERIES  MOTOR. 

AT  present,  nearly  all  electric  railways  in  this  country  use  the 
direct-current  series  motor  as  their  motive  power.  The  distribution 
of  the  electric  power  from  the  generating  station  to  the  car  motor 
is  inefficient  because  of  the  many  steps  involved.  The  power  is 
generated  as  alternating  current,  goes  through  step-up  transformers 
at  the  station  to  the  high-tension  transmission  line  and  to  the 
substation,  through  step-down  transformers,  changed  to  direct- 
current  power  by  a  rotary  converter,  and  is  then  conveyed  over 
the  D.C.  feeders  and  trolley  wire  to  the  car.  Probably  not  more 
than  60  per  cent  of  the  energy  generated  at  the  main  station 
reaches  the  car. 

In  case  the  alternating-current  series  motor  is  used  the  power 
distribution  is  much  simpler.  Only  a  step-up  transformer, 
feeder  and  trolley  wire  and  step-down  transformer  are  used  in  the 
distributing  system  so  that  the  distribution  is  accomplished 
much  more  efficiently,  both  as  regards  loss  of  power  and  main- 
tenance of  apparatus,  than  with  the  D.C.  system.  Probably 
if  the  A.C.  series  motor  was  as  efficient  and  reliable  as  the  D.C. 
series  motors  all  railway  installations  would  be  A.C. 

A  D.C.  series  motor  will  run  in  the  same  direction  whichever 
way  it  is  connected  to  the  line,  provided  the  relative  connection 
of  its  armature  and  field  is  undisturbed.  It  follows  that  a  D.C. 
series  motor  would  exert  a  unidirectional  torque  if  connected  to 
an  A.C.  line.  Such  an  application  of  the  D.C.  motor  is  not 
feasible  because  of  the  high  impedance  of  its  field  coils  and  be- 
cause of  the  fact  that  its  solid  iron  poles  and  yoke  would  not 
sufficiently  reverse  their  polarity  with  the  frequency  of  the  A.C. 
supply.  There  would  also  be  heavy  sparking  at  the  brushes 
when  the  motor  was  running. 

It  is  evident  that  the  field  frame,  as  well  as  the  armature  of 
an  A.C.  series  motor  must  be  of  laminated  iron  because  of  the 
continually  reversing  field  flux.  Because  of  this  fact  the  A.C. 
motor  cannot  be  as  cheaply  constructed  nor  of  such  rigid  me- 
chanical design  as  the  D.C.  motor. 

216 


THE  SINGLE-PHASE  SERIES   MOTOR 


217 


But  even  when  the  field  frame  is  laminated  there  still  remain 
two  series  defects  to  overcome,  namely  low  power  factor  and 
poor  commutation. 

The  reason  for  the  low  power  factor  may  be  easily  seen  by 
considering  the  action  of  the  field  and  armature  windings.  In 
Fig.  132  *  is  represented  the  elemen- 
tary series  A.C.  motor.  The  field 
coil  produces  useful  flux,  which  is 
necessary.  The  armature  produces 
a  vertical  flux  (Fig.  132),  which  is  of 
no  use  in  the  operation  of  the  motor 
but  helps  to  produce  a  low  power 
factor.  By  putting  a  compensating 
winding  around  the  armature,  con- 
taining the  same  number  of  turns  as  the  armature,  but  so  con- 
nected as  to  produce  a  M.M.F.  opposite  to  that  of  the  armature, 
then  the  armature  and  compensating  coil  will  just  neutralize  and 
produce  no  magnetic  field  and  so  produce  no  lag  in  the  current. 
Such  a  compensating  winding  is  illustrated  in  Fig.  133,  the  com- 
pensating winding  being  conductively  connected  to  the  rest  of  the 


FIG.  132. 


FIG.  133. 


FIG.  134. 


windings.  As  the  useless  armature  flux  is  alternating,  the  com- 
pensating coil  might  simply  be  short-circuited  as  in  Fig.  134,  or 
the  compensating  coil  may  be  put  in  series  with  the  main  field 
and  the  brushes  short-circuited.  In  these  two  schemes  the  arma- 
ture flux  is  neutralized  by  magnetic  coupling  of  armature  and 
compensating  coil,  not  electrical  coupling  as  in  Fig.  133. 

The  next  step  in  improving  the  power  factor  is  to  make  the 
number  of  turns  in  the  field  as  small  as  possible,  using  a  corre- 
spondingly greater  number  on  the  armature.  This  large  number 

*  Of  course  the  A.C.  series  motor  does  not  have  actual  projecting  pole 
pieces  as  shown  in  Figs.  132-135.  The  field  winding  and  field  core  of  a  series 
A.C.  motor  resemble  in  appearance  those  of  an  induction  motor. 


218 


ALTERNATING  CURRENTS 


of  armature  turns  does  not  produce  a  low  power  factor  as  the 
armature  winding  is  compensated. 

The  next  thing  to  consider  is  the  commutation  of  a  series  A.C. 
motor.  This  is  by  far  the  most  difficult  point  for  the  designer  to 
satisfactorily  solve.  The  armature  winding  of  the  A.C.  motor  is 
nearly  the  same  as  that  of  the  D.C.  motor  so  it  might  appear  at 
first  that  if  the  coil  undergoing  commutation  were  in  the  neutral 
plane  of  the  field,  commutation  would  be  no  more  difficult  than 
with  the  D.C.  motor. 

There  is  in  the  A.C.  series  motor  armature  not  only  an  E.M.F. 
produced  by  the  rotation  of  the  armature  through  the  field  flux  but 
also  an  E.M.F.  due  to  the  time  rate  of  change  of  the  field  flux,  this 
last  E.M.F.  being  due  to  transformer  action  of  the  field  coils  and 

armature  coils.  It  will  also  be  seen  from 
Fig.  135  that  coils  A  A,  those  being  corn- 
mutated,  are  in  the  position  where  the 
transformer  action  has  its  maximum  ef- 
fect, while  in  coils  BB  the  effect  is  zero. 
The  three  factors  affecting  this  trans- 
former E.M.F.  are,  the  number  of  arma- 
ture turns  in  series  between  commutator 
bars,  the  frequency  of  the  power  supply 

and  the  density  at  which  the  field  is  worked.     Decreasing  any  or 
all  of  the  quantities  tends  to  improve  commutation.  The  armature 
of  A.C.  series  motors  are  generally 
made  with  only  one  or  two  turns 
between     commutator     bars,     are 
operated  on  circuits  of  25  cycles  or 
less  and  the  field  densities  employed 
are  less  than  those  used  with  D.C. 
motors. 

The  transformer  E.M.F.  in  the 
coil  undergoing  commutation  pro- 
duces in  this  coil  a  large  short-circuit 
current,  the  coil  being  short-cir- 
cuited through  the  brush.  It  is 
the  breaking  of  this  current  as  the 

coil  moves  from  under  the  brush  which  produces  in  this  type 
of  motor  the  excessive  sparking.  One  method  of  limiting  this 
short-circuit  current  is  by  what  are  called  "  preventive  leads," 
being  wires  of  comparatively  high  resistance  connecting  the 


FIG.  136. 


THE  SINGLE-PHASE  SERIES  MOTOR  219 

coils  to  the  commutator  bars,  as  shown  in  Fig.  136,  at  r,  r,  etc. 
Suppose  the  preventive  lead  has  a  resistance  equal  to  twice  that  of 
the  coil  itself.  Then  when  the  coil  is  short-circuited  through  the 
brush  the  short-circuit  current  is  one-fifth  as  large  as  it  would  be 
without  the  resistance  leads.  It  might  seem  that  these  leads 
would  so  increase  the  armature  resistance  that  low  efficiency 
would  result.  This  is  not  the  case.  If  there  are  fifty  coils  be- 
tween adjacent  brushes  the  armature  resistance  with  leads  is 
only  f£  of  what  it  would  be  without  the  preventive  leads.  The 
resistance  of  the  leads  is  large  compared  to  the  resistance  of  one 
coil,  but  low  compared  to  the  resistance  of  the  whole  armature. 

The  torque  of  the  A.C.  motor  is  pulsating,  having  a  frequency 
twice  that  of  the  power  supply;  the  entire  field  frame  is  subject 
to  core  losses;  the  short-circuit  coil  is  subject  to  PR  losses  pro- 
duced by  the  transformer  E.M.F.;  the  power  factor  is  always 
less  than  one;  all  of  these  effects  result  in  less  horse  power  per 
pound  of  material  than  is  obtained  from  the  D.C.  series  motor. 
Also  the  commutator  of  the  A.C.  motor  is  likely  to  require  more 
attention  than  that  of  the  D.C.  motor. 

The  speed  load  curve  of  the  A.C.  motor  is  practically  the 
same  as  that  of  the  D.C.  motor.  The  power  factor  of  the  motor 
increases  with  speed  for  two  reasons.  The  reactance  drop  in  the 
field  winding  decreases  with  the  decrease  of  current  at  increased 
speed  and  the  C.E.M.F.  of  rotation,  which  is  an  energy  or  dis- 
sipative  reaction,  in  phase  with  the  current,  increases  with  the 
speed. 

A  vector  diagram  of  the  different  reacting  E.M.F.'s  in  the  single- 
phase  series  motor  serves  well  to  illustrate  the  effect  of  the  com- 
pensating winding,  etc.  In  Fig.  137  the  radius  of  the  circular 
arc  is  taken  equal  to  the  E.M.F.  impressed  on  the  motor.  In 
Fig.  137  we  have 

OA  =  Full-load  field  reactance  drop. 
AB  =  Full-load  armature  reactance  drop. 
OC  =  Full-load  armature  and  field  resistance  drop. 
CD  =  Full-load  compensating  winding  resistance  drop. 
OE  =  Full-load  current  motor  impedance  drop,  uncompensated. 
EF  =  Full-load  current  motor  C.E.M.F.  of  rotation,  uncom- 
pensated. 

cos  <£  =  Full-load  current  motor  power  factor,  uncompensated. 
OG  =  Full-load  current  motor  impedance  drop,  compensated. 


220 


ALTERNATING  CURRENTS 


GH  =  Full-load  current  motor   C.E.M.F.  of  rotation,  com- 
pensated, 
cos  0'  =  Full-load  current  motor  power  factor,  compensated. 

This  diagram  shows  that  the  power  factor  is  considerably 
improved  by  the  addition  of  the  compensating  winding.  With 
compensation,  of  course,  the  only  reactance  drop  occurring  in  the 
motor  is  OA,  that  due  to  the  field  winding.  It  might  seem  that 

the  efficiency  of  the  motor  would 
be  decreased  with  compensation 
owing  to  the  extra  IR  drop  shown 
at  CD.  The  efficiency  is,  however, 
greater  in  the  compensated  motor 
than  in  theuncompensatedone,  be- 
cause for  a  given  output  more  cur- 
rent is  required  in  the  latter  type, 
due  to  its  lower  power  factor,  and 
also  the  core  loss  is  reduced.  For 
a  given  current  the  speed  is  also 
considerably  higher  in  the  com- 
pensated motor;  this  is  due  to  the 
decreased  reactance  drop,  thereby 
leaving  a  larger  percentage  of  the 

impressed  E.M.F.  to  be  used  up  in  C.E.M.F.  of  rotation,  and  as 
the  field  is  assumed  the  same  strength  in  both  cases  the  compen- 
sated motor  must  run  at  a  higher  speed  than  the  uncompensated 
type. 

Make  a  test  upon  a  series  A.C.  motor  (using  brake  for  load), 
reading  volts,  amperes,  watts  input,  torque  and  r.p.m.  Take 
readings  in  about  ten  steps  between  the  load  which  gives  150 
per  cent  rated  speed  and  that  which  gives  150  per  cent  rated 
current,  the  voltage  and  frequency  being  kept  constant  at  their 
rated  values. 

Make  a  similar  run  with  the  compensating  winding  connected 
in  the  motor  circuit  but  reversed,  so  that  its  inductance  adds  to 
that  of  the  armature  instead  of  serving  to  neutralize  it.  Note 
the  commutation  in  these  two  runs. 

Make  a  run,  having  the  compensating  winding  properly  con- 
nected, using  for  power  supply  some  frequency  widely  differ- 
ent from  the  rated  value  (e.g.,  run  a  25-cycle  motor  on  a 
60-cycle  line).  Note  and  record  the  commutation  as  compared 
with  the  previous  run.  Make  a  similar  run  using  the  motor 


FIG.  137. 


THE  SINGLE-PHASE  SERIES  MOTOR  221 

on  a  D.C.  line  of  the  same  voltage  as  the  motor  rating.  As 
the  losses  will  be  less  in  this  case  than  when  the  motor  is  being 
run  on  A.C.  the  full-load  current  may  be  reckoned  as  110  per 
cent  of  the  A.C.  rating. 

With  normal  frequency  and  various  voltages  (beginning  with 
low  values)  and  the  armature  locked,  read  watts,  amperes  and  volts 
input  at  about  six  points  from  zero  to  50  per  cent  or  75  per  cent 
over-load  current.  Take  care  that  the  motor  does  not  over- 
heat with  the  higher  values  of  current.  Also  read  amperes, 
watts  and  starting  torque  with  full-load  current,  both  D.C.  and 
A.C.  Calculate  amperes  and  watts  per  pound-foot  of  starting 
torque  for  A.C.  and  D.C. 

With  current  input  as  abscissa,  draw  four  sets  of  curves,  one 
set  for  each  run.  In  each  set,  plot,  against  current  input,  speed, 
torque,  watts  input,  current  input,  efficiency  and  power  factor. 
From  the  "  locked-saturation  "  curve  and  one  of  the  readings 
from  the  first  test  construct  the  circle  diagram*  and  predict  the 
various  characteristics  of  the  motor. 

What  conclusions  can  be  drawn  from  the  starting  torque 
characteristics  on  A.C.  and  D.C.  power? 

Note.  —  Care  must  be  exercised  that  the  motor  speed  does  not  rise  to  a 
dangerous  value.  A  switch  should  be  so  placed  that  the  man  taking  speed 
can  immediately  open  the  A.C.  supply  circuit  if  the  speed  exceeds  the  safe 
limit  (5000-6000  ft./min.  =  maximum  peripheral  speed). 

*  For  construction  of  this  diagram  see  Crocker  and  Arendt,  "  Electric 
Motors,"  page  249. 


EXPERIMENT   XXXIX. 
THE  MERCURY  ARC  RECTIFIER. 

Two  electrodes  in  a  vessel  which  has  been  exhausted  to  a  high 
degree  of  vacuum  are  practically  insulated  from  one  another  for 
ordinary  voltages.  The  phenomena  which  result  from  a  gradu- 
ally increasing  potential  difference  of  the  electrodes  are  of  too 
involved  and  complex  a  nature  to  be  fully  analyzed  here,  but  it 
is  believed  that  this  subject  of  conduction  of  electricity  by  gases 
is  of  enough  importance  to  be  seriously  studied  by  the  electrical 
engineer.  It  is  undoubtedly  true  that  a  thorough  comprehension 
of  the  novel  effects  produced  in  high  tension  transmission  of 
electrical  power  can  only  be  obtained  by  a  study  and  application 
of  the  electron  theory.* 

In  the  analysis  of  the  mercury  arc  rectifier  only  those  funda- 
mental facts  which  are  necessary  for  an  explanation  of  the  action 
of  the  rectifier  will  be  taken  up  and  briefly  treated.  If  a  tube 
in  which  two  electrodes  are  sealed  is  exhausted  to  a  high  degree 
of  vacuum  and  an  increasing  voltage  is  impressed  on  the  elec- 
trodes it  will  be  found  that  a  very  small  current  will  pass  through 
the  tube,  perhaps  one  or  two  milliamperes.  The  gas  in  the  tube 
is  said  to  be  un-ionized  and  in  this  state  has  a  very  high  resistance. 
As  the  potential  difference  of  the  electrodes  is  increased  the 
current  will  gradually  rise,  but  until  a  certain  critical  voltage  is 
reached  will  not  exceed  a  few  milliamperes. 

When  the  potential  difference  reaches  a  value  of  perhaps  2500 
volts  the  current  will  suddenly  rise  and,  if  the  potential  dif- 
ference is  maintained,  the  current  will  be  of  such  a  value  that  in 
a  few  seconds  the  negative  electrode  will  melt.  This  sudden 
rise  in  current  is  explained  by  saying  that  when  the  potential 
gradient  in  the  tube  reaches  a  certain  value  the  gas  becomes 
ionized  (so-called  "  ionization  by  impact  ")  and  when  ionized 
to  a  considerable  extent  a  gas  becomes  a  fair  conductor. 

The   potential   drop   is  greatest  near  the   cathode,   in  fact, 

*  The  student  is  referred  to  J.  J.  Thomson's  "  Conduction  of  Electricity 
through  Gases,"  for  a  complete  exposition  of  this  theory. 

222 


THE   MERCURY  RECTIFIER  223 

practically  all  of  the  E.M.F.  applied  to  the  tube  is  used  up  in 
overcoming  the  resistance  from  the  cathode  to  the  adjacent  gas. 
This  potential  difference  is  called  the  "  cathode  drop,"  and  this 
drop  is  very  high  when  such  an  electrode  as  iron  is  used. 

We  have  now  the  fact  that  an  ionized  gas  is  a  conductor,  but 
that  an  excessive  voltage  is  necessary  to  overcome  the  cathode 
drop.  As  the  power  used  in  the  circuit  is  equal  to  the  product 
of  volts  X  amperes,  it  is  at  once  evident  why  the  negative  elec- 
trode gets  so  hot. 

The  next  thing  to  consider  is  the  action  of  such  a  tube  when 
one  electrode  is  made  of  mercury,  the  gas  in  the  tube  is  mercury 
vapor,  and  the  other  electrode  is  some  such  substance  as  iron. 

Suppose  a  tube  made  as  shown  in  Fig.  138.  The  voltage  of 
the  generator  may  be  brought  to  a  very  high  value  but  still 
no  appreciable  current  will 
pass  through  the  tube  be- 
cause the  enclosed  gas  is  not 
ionized.  If  now,  by  means 
of  the  auxiliary  electrode  C 
and  induction  coil  H,  a 
spark  is  made  to  pass  in- 
to the  mercury  electrode, 
immediately  ions  are  freed 
from  the  mercury,  the  gas  is  ionized  and  a  current  of  a  value  of 
several  amperes  (limited  by  resistance  R)  will  flow  from  B  to  A. 
Also  the  potential  difference  between  B  and  A,  even  when  10  or 
20  amperes  are  flowing,  is  only  about  15  volts. 

If,  however,  the  generator  terminals  are  reversed,  making  B 
the  cathode,  it  makes  no  difference  how  well  the  gas  is  ionized, 
no  appreciable  current  will  flow  through  the  tube  unless  an 
excessively  high  potential  difference  is  used.  We  have  here  then 
the  secret  of  the  mercury  arc  rectifier,  i.e.,  selective  electrodes. 
When  the  mercury  is  used  as  cathode,  the  cathode  drop  is  about 
4  volts;  when  the  iron  is  used  as  cathode  the  cathode  drop  is 
perhaps  2500  volts. 

If  such  a  tube  is  connected  to  a  source  of  A.C.  power  and  the 
gas  in  the  tube  is  ionized,  whenever  the  iron  electrode  is  15  volts 
or  more  higher  in  potential  than  the  mercury  electrode,  current 
will  flow.  If  the  tube  is  connected  as  shown  in  Fig.  139  and  an 
oscillogram  is  taken  of  the  current  and  voltage  in  the  circuit  they 
would  be  as  given  in  Fig.  140.  Such  a  tube  would,  therefore,  be 


224 


ALTERNATING  CURRENTS 


giving  pulses  of  unidirectional  current.  The  drop  of  15  volts 
in  the  tube  is  nearly  independent  of  the  current  flowing,  so  that 
the  amplitude  of  the  current  pulses  depends  only  on  the  value 
of  the  resistance  R.  The  ionizing  spark  must  continually  be 
operating,  otherwise  current  would  not  start  to  flow  through  the 

tube  until  excessively 
high  value  of  potential 
difference  was  used. 
After  the  current  starts 
to  flow,  however,  the 
ionizing  spark  may  be 
cut  off  and  the  current 

FIG.  139.  itself  will  maintain  the 

ionization.  If  the  cur- 
rent stops  for  a  fraction  of  a  second  (something  less  than  0.000001 
second)  the  vapor  recovers  its  insulating  quality  and  the  ionizing 
spark  must  be  again  used  to  start  the  current  flowing. 

This  recovery  of  insulating  power  is  due  to  the  fact  that  imme- 
diately the  current  stops  flowing,  the  ions  disappear,  some  going 
to  the  sides  of  the  tube  by  dispersion  (and  clinging  to  the  sides  of 


1=0 


I  \  Iron  Negative 
1=0 

Zero  Value  of  I 


1=0 


Zero  Line  off  E.M.F.         \       &  Yolts 


7 


FIG.  140. 

the  tube)  and  the  rest  recombining.  As  the  recombination  is  not 
very  rapid  at  low  pressure  it  is  probable  that  dispersion  accounts 
for  most  of  the  effect. 

The  next  step  in  the  development  of  the  rectifier  is  to  so  con- 
struct it  that  both  alternations  may  pass  through  the  tube,  both 
going  through  the  tube  in  the  same  direction.  This  is  done  by 
constructing  a  tube  with  one  mercury  electrode  and  two  iron 
electrodes,  as  shown  in  Fig.  141.  The  tube  is  connected  to  the 
line  as  shown.  The  two  sides  of  the  A.C.  line  are  connected  to 
the  two  anodes  B  and  Z),  and  the  cathode  is  connected  to  the 
junction  of  the  two  reactance  coils  F  and  G.  Between  A  and  E 


THE  MERCURY  RECTIFIER 


225 


are  connected  some  storage  ,cells  which  may  be  charged  from  the 
tube. 

The  operation  is  now  as  follows:  the  gas  being  supposed 
ionized  continually  (by  electrode  C  and  spark  if  necessary). 
When  B  is  positive  with  re- 
spect to  A  (and  more  than 
15  volts  above  A  in  poten- 
tial) current  will  flow  from 
B  to  A,  through  the  battery 
K,  through  F  and  so  back 
to  the  A.C.  line.  During 
the  next  alternation  D  be- 
comes positive  with  respect 
to  A,  so  current  flows  from 
D  to  A,  through  K  and  G 
and  so  back  to  the  line. 
Both  pulses  of  current  pass 
through  K  in  the  same  di- 
rection, so  that  this  tube 
will  give  a  unidirectional 
pulsating  current  as  shown 
in  Fig.  142.  As  the  ioniza- 

tion,  produced  by  the  current  itself,  would  disappear  once  every 
alternation  (at  the  points  of  zero  current  in  Fig.  142),  the  gas 
must  be  kept  continually  ionized  by  the  coil  H  and  electrode  C. 
As  this  is  inconvenient,  a  method  is  sought  to  overcome  this 
difficulty. 

So  long  as  current  is  flowing  into  the  cathode  the  gas  remains 
ionized  and  it  matters  not  from  which  anode  the  current  is 

coming.  If,  therefore,  the 
current  can  be  made  to  start 
from  D  before  the  current 
from  B  has  reached  zero 

value,  the  tube  will  keep  itself  continually  ionized  and  the  elec- 
trode C  need  be  used  only  for  starting. 

This  is  the  condition  which  actually  obtains  in  the  tube  and 
it  is  brought  about  by  the  reactance  coils  F  and  G.  In  Fig.  143 
is  drawn  the  wave  of  potential  difference  between  B  and  A .  As 
soon  as  a  value  of  (15  volts  +  C.E.M.F.  of  battery)  is  reached 
current  begins  to  flow  at  a,  Fig.  143,  and  follows  practically 
sine  wave  as  shown  by  the  dashed  line.  If  the  reactance  coil  did 


FIG.  142. 


226 


ALTERNATING  CURRENTS 


not  come  into  play  the  current  wave  would  continue  as  shown  by 
the  dashed  line  and  would  reach  zero  value  at  time  b.  The 
current  from  anode  D  will  not  begin  until  time  c,  so  that  were  it 


FIG.  143. 

not  for  the  reactance  coils  during  the  time  (6-c)  there  would  be 
no  current  in  the  tube  and  the  gas  would  lose  its  ionization. 

During  the  time  (a-e)  the  reactance  coil  G  is  storing  magnetic 
energy  and  as  the  current  starts  to  decrease  at  time  e,  the  re- 
actance coil  begins  to  discharge  its  energy  and  tends  to  sustain 
the  current.  This  results  in  a  change  in  the  current  wave  form 
as  shown  in  Fig.  144,  where  the  E.M.F.  waves  and  current  waves 


Vl5yolts+C.E.M.F. 


FIG.  144. 

of  both  anodes  are  shown.  The  two  current  waves  now  lap  over 
one  another  by  several  degrees  so  that  the  current  wave  through 
the  cathode  is  shown  as  in  Fig.  145.  Hence,  a  tube  constructed 
with  two  anodes  and  used  with  reactance  coils  will  give  through 
the  cathode  a  pulsating  unidirectional  continuous  current.  The 
variation  in  amplitude  may  be  30  per  cent  or  40  per  cent  of  the 
maximum  value. 

As  a  storage  cell  becomes  charged  its  C. E.M.F.  rises,  and, 
therefore,  arrangement  must  be  made  whereby  the  rectifier  volt- 
age may  be  increased  as  the  charge  continues.  Also  different 


THE  MERCURY  RECTIFIER 


227 


numbers  of  cells  will  be  charged  at  different  times,  so  that  here 
also  different  voltages  are  desired.     The  rectifier  as  actually 


Time 
FIG.  145. 

installed  permits  an  adjustment  of  the  voltage  before  starting 
the  charge  and  then  a  variation  in  the  voltage  as  the  charge 
continues.  A  variable  reactance 
is  placed  in  the  A.C.  supply  line, 
which  reactance  is  varied  with 
time  of  charging.  In  place  of 
the  transformer  and  reactance 
coils  an  adjustable  auto-trans- 
former is  used,  as  shown  in  Fig. 
146.  In  this  figure  the  starting 
arrangement  is  also  shown.  In- 
stead of  using  a  spark  from  an 
induction  coil  to  start  the  ioniza- 
tion,  an  arc  is  used.  With  switch 
Si  closed  the  tube  is  tipped  so 
that  mercury  connects  the  two 
electrodes  A  and  C,  and  a  current 
flows,  the  amplitude  being  limited 
by  resistance  Ri.  When  the  tube 
is  tipped  back  an  arc  is  formed 
as  the  bridge  of  mercury  is  broken, 
the  gas  is  ionized  and  if  switch  S2, 
connecting  in  the  starting  resist- 
ance Rz,  is  closed,  the  tube  will 

begin  to  operate,  using  R2  as  load.  When  the  tube  has  been 
running  a  few  seconds  through  R2}  to  get  warmed  up,  £3  may  be 
closed  and  S2  opened.  Si  should  be  opened  as  soon  as  the  tube 
is  operating. 

The  D.C.  voltage  available  at  the  load  is  about  25  per  cent  of 
the  A.C.  voltage  between  the  two  anodes  D  and  B.     In  starting 


.C.LiueO-» 

FIG.  146. 


228 


ALTERNATING  CURRENTS 


to  charge,  the  contacts  MM  of  the  autotransformer  are  set  at  the 
proper  tap  with  the  reactance  coil  X,  all  in  the  circuit.  Then 
as  the  charge  progresses  the  reactance  may  be  gradually  cut  out 
if  it  is  desired  to  maintain  the  charging  current  constant. 

As  considerable  power  is  used  in  the  tube  itself,  the  upper  part 
H  is  made  comparatively  large  to  give  the  required  radiating 
surface.  The  mercury  boils  and  vaporizes  from  the  cathode,  is 
condensed  on  the  walls  of  the  tube  and  serves  well  to  distribute 
the  heat  uniformly  over  the  surface  of  the  tube. 

The  efficiency  of  an  arc  rectifier  is  nearly  independent  of  load 
(the  efficiency  of  the  tube  itself  being  strictly  so),  but  depends 


Voltage  Curves. 

Mercury  Arc  Rectifier 

All  voltages  to  game  Scale 


FIG.  147. 

upon  the  value  of  D.C.  voltage  at  the  load,  the  efficiency  in- 
creasing with  the  voltage. 

The  power  factor  of  the  rectifier  depends  somewhat  upon  the 
load  and  also  upon  how  much  of  the  reactance  coil  X  is  being 
used,  but  generally  it  is  well  up  towards  85  per  cent  or  90  per 
cent. 

The  rectifier  has  an  inherent  regulation  of  about  20  per  cent 
from  rated  load  of  the  tube  to  the  minimum  load  the  tube  will 
carry. 

The  wave  forms  of  the  current  and  E.M.F.'s  in  the  different 
parts  of  the  rectifier  circuit  are  to  be  obtained  either  by  ondo- 
graph  or  oscillograph.  The  waves  to  be  taken  are  A.C.  line  volt- 
age and  current;  voltage  between  each  anode  and  cathode; 


THE  MERCURY  RECTIFIER 


229 


current  furnished  by  each  .anode;  D.C.  current  (cathode  cur- 
rent), and  voltage  across  load.  All  of  these  curves  must  be  taken 
with  the  same  setting  of  the  curve-tracing  device  so  that  their 
proper  phase  relations  are  obtained;  sample  curves  from  a  10- 
ampere  220-volt  tube  are  shown  in  Fig.  147  and  Fig.  148;  they 
were  taken  by  means  of  the  ondograph. 

Now  insert  a  high  inductance  in  the  load  line  (a  transformer 
coil,  e.g.)  and  take  another  curve  of  D.C.  current  to  see  if  pulsa- 
tions are  reduced. 

Make  runs  to  obtain  efficiency,  power  factor  and  regulation 
as  follows : 

Reactance  X  all  on,  readings  of  A.C.  line  volts,  current  and 
watts,  D.C.  current  and  volts,  for  J,  \,  f,  full  and  1J  rated 


A.C.  &  D.C.  Current  Curves 

Mercury  Arc  Rectifier 
All  Currents  to  same  Scale 


FIG.  148. 

current.  Run  to  be  made  with  autotransformer  set  to  give 
highest  D.C.  volts. 

Similar  run  with  autotransformer  set  to  give  minimum  D.C. 
volts. 

Take  similar  runs  with  reactance  all  cut  out. 

For  all  runs  plot  efficiency,  power  factor  and  D.C.  volts 
against  D.C.  load  current  as  abscissa. 

Note.  —  In  measuring  a  pulsating  unidirectional  current  or  voltage,  A.C. 
instruments  should  be  used,  taking  reversed  readings  to  get  rid  of  error  caused 
by  stray  field.  D.C.  instruments  will  not  give  accurate  indications  when  the 
pulsations  have  an  appreciable  magnitude.  Why? 


APPENDIX. 

ILLUSTRATIONS  OF  THE  USE  OF  THE  ONDOGRAPH  AND  OSCIL- 
LOGRAPH FOR  SOLVING  SOME   OF  THE  MORE  INVOLVED 
QUESTIONS  ENCOUNTERED   IN  A  COURSE   ON 
ALTERNATING-CURRENT  TESTING. 

AN  inductance  of  .11  henry  having  a  resistance  of  1.4  ohms  was 
connected  through  a  resistance  of  .6  ohm  to  a  condenser  of  80 
microfarads  having  an  inappreciable  series  resistance.  A  switch 
was  placed  in  the  circuit  so  that  the  connection  diagram  was  as 
shown  in  Fig.  18  of  the  text.  The  condenser  was  charged  to  100 
volts  difference  of  potential  and  then  the  switch  was  closed, 
allowing  the  condenser  to  discharge  through  the  inductance  and 
resistance.  The  oscillatory  current  in  the  circuit  is  shown  by 


,V\/Wvww 


curve  1,  Plate  1.     The  frequency  of  the  discharge  by  the  formula, 
/  =  ^—  V/  Y7i)  (resistance  neglected)  was  calculated  to  be  53.5  cycles 

Z  7T    '     Lt\j 

per  second  while  the  film  measured  about  54  cycles  per  second. 

The  resistance  of  the  circuit  was  then  increased  to  about  8  ohms 
and,  when  the  condenser  was  again  charged  to  100  volts,  and  dis- 
charged through  the  inductance,  curve  2,  Plate  1,  was  obtained. 
The  very  rapid  damping,  as  compared  to  curve  1,  is  at  once  seen. 
The  change  in  the  period  of  the  current  due  to  the  increased  resist- 
ance is  not  measurable  on  the  film.  The  resistance  of  the  circuit 

231 


232 


APPENDIX 


was  then  increased  to  about  150  ohms  and  curve  1,  Plate  2,  was  ob- 
tained, to  the  same  scale  as  the  curves  of  Plate  1.     Upon  changing 


the  resistance  of  the  oscillograph  circuit  curve  2,  Plate  2,  was  ob- 
tained as  the  form  of  the  discharge  of  the  condenser.  Evidently 
there  was  no  oscillatory  character  to  the  discharge  current  in  this 
circuit.  Upon  calculation  it  will  be  seen  that  with  150  ohms  in 

R2  1 

the  circuit  the  term  -j-y-2  is  greater  than  the  term  j-~  and,  by  in- 
*±  Lr  LJ\J 

spection  of  the  equation  on  page  38  of  the  text,  it  is  evident  that 
no  oscillation  will  take  place  under  such  a  condition. 


An  inductance  of  about  .005  henry  was  placed  in  series  with  a 
resistance  of  about  10  ohms  and  a  condenser  of  80  microfarads  and 


APPENDIX 


233 


was  switched  to  a  110-volt  60-cycle  line.     The  peculiar  shape  of 
the  starting  current  is  seen  by  reference  to  Plates  3  and  4.     Of 


course  in  the  circuit  used  the  damping  factor  was  very  large,  as 
the  inductance  was  of  such  a  small  value.  The  current  reached 
its  steady  state  in  this  circuit  in  about  two  cycles. 


•4-,-/- 


.--  .-f 


The  curves  of  Plate  5,  which  were  taken  by  the  ondograph,  show 
the  effect  of  the  armature  reaction  upon  the  field  of  a  single  phase 
alternator.  The  curve  (A)  shows  the  voltage  induced  in  the  series 
field  of  the  machine  and  curve  (B)  gives  the  armature  E.M.F.  It  is 


234  APPENDIX 

seen  that  curve  (A)  has  twice  the  frequency  of  curve  (B).  The 
machine  used  was  designed  as  a  rotary  converter;  hence,  the  pres- 
ence of  the  series-field  winding.  It  was  used  instead  of  winding 
an  additional  search  coil  on  the  pole  face.  When  the  machine  was 
loaded  two  phase  there  was  no  perceptible  E.M.F.  induced  in  the 
series-field  winding,  showing  that  the  balanced  polyphase  load 
produced  no  pulsations  in  the  field  strength. 


Two  alternators  of  different  design  were  being  operated  in 
parallel.  It  was  impossible  to  reduce  the  current  circulating  be- 
tween the  two  machines  to  less  than  3  amperes,  which  was  about 
15  per  cent  of  full-load  current.  A  low  resistance  was  put  in  the 
circuit  of  the  two  armatures  and  an  oscillograph  record  of  this 
circulating  current  was  taken  and  it  is  reproduced  in  Plate  6. 
The  line  E.M.F.  is  given  as  reference  curve.  It  may  be  seen  that 
this  circulating  current  is  very  complex  in  form,  so  that  it  could 
not  be  eliminated  by  variation  of  field  strength,  etc.  The  shape 
of  this  circulating  current  is  probably  due  to  the  fact  that  the  two 
alternators  used  had  a  different  number  of  armature  teeth  per 
pole.  Other  machines  tested  gave  even  more  complex  curves  for 
circulating  current. 

The  question,  as  to  whether  or  not  the  secondary  E.M.F.  of  a 
transformer  is  exactly  similar  in  form  to  the  impressed  E.M.F., 
was  raised  by  one  of  the  students.  It  seemed  possible  that  the 
effect  of  the  losses  in  the  transformer  core  might  be  such  that  slight 


APPENDIX  235 

irregularities  in  the  impressed  E.M.F.  might  be  eliminated  in  the 
secondary  E.M.F.  So  various  forms  of  wave  forms  were  impressed 
upon  a  transformer  having  quite  high  core  losses  but  the  results 
showed  that,  within  the  limits  of  accuracy  of  the  curve-tracing 


apparatus,  the  two  wave  forms  were  exactly  similar.  Plate  7 
shows  the  result  of  one  test;  the  impressed  E.M.F.  is  shown  at  E1} 
the  primary  current  at  /i,  and  the  secondary  E.M.F.  is  shown  by 
the  curve  E2. 

The  alternator  used  hi  this  test  is  the  one  referred  to  in  the  dis- 
cussion of  Fig.  87  of  the  text.  A  very  pronounced  eleventh  har- 
monic is  present  in  the  wave  form. 


Plate  8.   The  excessive  current  taken  by  a  transformer  on  being 
switched  to  a  line  of  normal  voltage  is  well  shown  on  this  oscillo- 


236 


APPENDIX 


gram.  It  is  seen  that  the  large  value  of  current  lasts  for  only  a 
very  few  cycles  and  that  the  magnitude  of  the  current  at  starting 
may  vary  greatly.  On  Plate  8  the  magnitude  of  current,  when 
the  switch  was  closed  the  first  time,  is  about  ten  times  normal 
magnetizing  current  and  the  next  time  the  switch  was  closed  the 
first  rush  of  current  gives  only  about  six  times  normal  current. 

It  is  quite  likely  that  if  the  times  of  disconnecting  the  trans- 
former and  of  switching  it  on  the  line  again  should  happen  to  be 
the  worst  possible,  as  described  in  the  text  in  connection  with 
Figs.  59  and  60,  that  the  starting  current  would  be  several  times 
the  full-load  current  of  the  transformer. 


Plate  9.  The  idea  of  the  "wabbling"  neutral  is  well  brought  out 
in  this  ondograph  record.  The  three  transformers  were  connected 
Y  —  A,  and  the  delta  was  left  open.  The  ondograph  traced  the 
curve  of  voltage  between  the  center  of  the  Y  (the  "wabbling" 
neutral)  and  an  artificial  neutral  obtained  by  connecting  to  the 
line  feeding  the  transformers  three  noninductive  resistances  con- 
nected in  Y.  The  line  E.M.F.  wave  is  drawn  for  reference.  The 
magnitude  of  the  third  harmonic  was  about  18  per  cent  that  of 
the  line  E.M.F. 

The  open  delta  of  Plate  9  was  closed  through  a  low  resistance  and 
the  "wabbling"  neutral  became  stationary,  the  ondograph  show- 
ing no  difference  between  the  transformer  neutral  and  the  arti- 
ficial neutral.  There  was  a  third  harmonic  current,  however, 
flowing  around  the  closed  delta  and  this  is  shown  on  Plate  10,  as  is 
the  secondary  E.M.F.  wave  for  reference.  The  current  circulating 


APPENDIX 


237 


in  the  delta  was  evidently,  made  up  of  some  higher  harmonics 
besides  the  third,  as  its  amplitude  varies  in  a  series  of  "beats." 


The  set  of  curves  given  in  Plate  11  shows  the  voltage  relations  in 
three  transformers  connected  Y  —  A,  the  delta  being  open  and  the 
secondaries  unloaded.  The  line  voltage  (1)  is  nearly  a  sine  wave; 
the  form  of  the  voltage  wave  across  one  primary  coil  (2)  is  very 


much  distorted,  there  being  present  a  third  harmonic  of  consider- 
able magnitude ;  the  secondary  voltage  wave  (3)  is  exactly  similar 
to  that  of  the  primary,  and  across  the  open  point  of  the  delta  there 
is  a  large  voltage  consisting  for  the  most  part  of  third  harmonic  (4). 
This  curve  (4)  is  not  entirely  made  up  of  the  third  harmonic,  as 
may  be  seen  from  the  variation  in  its  amplitude.  The  voltage 


238 


APPENDIX 


across  the  open  delta  as  recorded  by  a  voltmeter  was  more  than 
half  as  large  as  the  normal  secondary  voltage.  In  this  case  the 
secondary  voltage  was  110  and  the  voltmeter  across  the  open  delta 
registered  72  volts.  When  the  delta  was  closed,  however,  the  cur- 
rent circulating  around  the  closed  delta  was  less  than  one  ampere. 


With  the  transformers  connected  Y  —  A  and  open  delta,  no 
load,  an  oscillogram  (Plate  12)  was  taken  of  line  voltage,  primary 
phase  voltage,  and  line  current.  The  two  voltage  curves  are  to  the 
same  scale,  the  primary  phase  voltage  consists  nearly  altogether  of 
fundamental  and  third  harmonic.  The  line  current  has  a  complex 
shape,  there  being  present  very  large  third  and  fifth  harmonics 
besides  the  fundamental. 

The  secondary  delta  was  closed,  but  with  no  load ;  the  quantities 
of  Plate  12  were  photographed  as  shown  in  Plate  13.  The  phase 
voltage  is,  as  nearly  as  can  be  detected,  a  pure  sine  wave  (with 
exception  of  the  fluctuations  same  as  in  line  E.M.F.,  due  to  teeth 
of  generator)  and  in  the  line  current  there  is  no  third  harmonic;  the 
fifth  harmonic,  however,  seems  to  be  as  large  as  in  Plate  12. 

The  transformers  were  connected  A  —  Y  and  not  loaded;  the 
primary  phase  current,  line  current  and  primary  E.M.F.  were  ob- 
tained as  given  in  Plate  14.  It  is  seen  that  in  such  a  connection 
the  primary  current  has  the  characteristic  form  of  a  magnetizing 
current;  in  the  line  current  there  is  again  a  very  pronounced  fifth 


APPENDIX 


239 


harmonic.  The  curves  described  in  Plates  9  to  14  are  for  un- 
loaded transformers;  when  load  is  put  on  the  transformers  these 
irregularities  nearly  disappear. 


It  has  been  shown  in  the  text  that  in  a  polyphase  synchronous 
machine  there  exists  a  nonpulsating  armature  reaction  which  will 
tend  to  concentrate  the  field  flux  in  one  tip  of  the  pole  or  the  other, 


240 


APPENDIX 


according  as  the  motor  is  under-  or  over-excited.  It  is  natural  to 
suppose  that  the  armature  current  under  such  conditions  may  not 
be  a  true  sine  wave  even  though  the  impressed  E.M.F.  be  a  pure 
sine  wave.  Some  current  curves  were  obtained  by  the  ondograph 
and  they  showed  that,  on  the  machine  tested,  a  very  noticeable 
distortion  of  the  current  form  took  place  as  the  field  strength  was 
varied  through  a  very  wide  range.  Curve  2  of  Plate  15  shows  the 
current  taken  by  the  motor  when  about  i  load  was  being  carried 
at  power  factor  equal  to  one;  even  at  this  power  factor  the  current 


form  was  distorted,  probably  due  to  the  fact  that  the  armature 
reaction  (the  machine  was  a  three-phase  one)  had  crowded  the 
field  flux  into  one  tip  of  the  field  poles.  Curve  1  gives  the  form  of 
the  line  E.M.F.  Curves  3  and  4  show  the  form  of  current  waves 
when  the  machine  was  carrying  the  same  load  as  before,  but  the  field 
was  first  over-excited  and  then  under-excited.  The  power  factor 
for  both  these  curves  was  approximately  .5.  It  will  be  noticed  that 
a  peculiar  distortion  takes  place  in  these  two  curves;  apparently 
there  is  a  large  third  harmonic  in  each  wave  but  the  phase  of  this 
third,  with  respect  to  the  fundamental,  is  different  in  the  two  cases. 
In  a  single  phase  rotary  the  armature  reaction  is  pulsating  and 
therefore  the  strength  of  the  magnetic  field  must  pulsate.  This 
being  the  case  one  would  expect  that  if  the  rotary  was  run  inverted 
the  counter  E.M.F.  generated  by  the  armature  would  not  be  a  con- 


APPENDIX 


241 


stant  quantity  and  that  the  current  input,  therefore,  would  not  be 
uniform  even  though  the  impressed  D.C.  voltage  were  constant. 
So  oscillograph  curves  were  taken,  showing  the  current  input  and 


current  output  of  a  small  single-phase  rotary.  On  one  machine 
tested,  (a  Crocker- Wheeler,  bipolar,  ring  wound,  32  coil  armature) 
the  field  was  not  stiff  and  the  armature  magneto-motive  force  was 
comparatively  large,  so  that  the  effect  of  armature  reaction  upon 


the  field  is  very  marked;  it  is  seen  in  Plate  16  that  the  D.C.  input 
pulsates  almost  200  per  cent  and  that  the  current  taken  from  the 
A.C.  end  is  very  far  from  being  a  sine  wave.  It  was  thought  at 
first  that  perhaps  the  angular  velocity  of  the  armature  was  vari- 
able, so  a  heavy  flywheel  was  mounted  on  the  armature  shaft,  but 


242  APPENDIX 

the  shape  of  the  two  curves  was  not  altered  by  this  addition.  In 
Plate  17  are  given  corresponding  curves  for  a  7  K.W.  Westinghouse 
4  pole  rotary,  which  had  a  low  armature  m.m.f.  and  a  compara- 
tively stiff  field.  It  is  seen  that  the  pulsations  in  the  D.C.  input 
are  very  much  less  marked  than  with  the  other  rotary. 


In  the  polyphase  rotary  the  armature  reaction  (when  there  is 
one)  is  constant  and  not  pulsating.  We  should,  therefore,  expect 
that  in  a  polyphase  rotary,  running  inverted  and  loaded  equally 
on  the  different  phases,  the  C.E.M.F.,  and  hence  the  current  input, 
would  be  constant.  Plate  18  gives  a  set  of  oscillograph  curves  to 
show  that  this  deduction  is  borne  out  by  results  actually  obtained 
from  such  a  machine. 


The  rotary  used  in  obtaining  the  curves  of  Plate  16  was  run 
from  the  A.C.  end,  and  the  curves  given  in  Plate  19  were  obtained. 
It  is  seen  that  the  effect  of  armature  reaction  does  not  affect  the 


APPENDIX 


243 


form  of  the  A.C.  current  input  to  a  very  marked  extent  but  that 
the  D.C.  voltage,  and  hence  the  current  output,  pulsate  with  double 
frequency.  The  machine  used  for  Plate  19  was  the  same  as  used 
for  obtaining  Plates  20-23. 


The  question  of  the  current  forms  in  the  different  coils  of  a  ro- 
tary converter  is  easily  solved  analytically  or  graphically,  but  it 
adds  somewhat  to  the  student's  confidence  in  his  results  if  the  curves 
are  actually  obtained  by  some  curve-tracing  apparatus  from  a 


machine  in  operation.  So  with  this  purpose  in  mind  we  fitted  up 
a  small  single-phase  rotary.  The  armature  consisted  of  32  coils 
and  the  field  was  bipolar.  Every  other  coil  of  the  armature  wind- 
ing was  opened  (on  the  end  opposite  the  commutator)  and  a  small 
amount  of  resistance,  wound  on  a  bobbin,  was  inserted  hi  the  coil 


244 


APPENDIX 


and  mounted  on  the  end  plate  of  the  armature.  An  extra  pair  of 
small  slip  rings  was  mounted  on  the  rotary  shaft  and  these  rings 
could  be  connected  across  any  one  of  the  inserted  resistance 
spools.  If  then  the  oscillograph  was  used  to  show  the  form  of  the 


IR  drop  across  these  different  spools  the  oscillogram  would  really 
show  the  current  form  in  that  coil  in  which  the  spool  in  question  was 
connected.  The  coils  in  which  the  resistance  spools  were  inserted 
were  numbered  consecutively,  1  being  in  a  coil  to  which  one  of  the 


A.C.  taps  of  the  rotary  was  connected.  Then  coil  2  is  really  the 
third  coil  from  the  A.C.  taps;  coil  3  is  really  the  fifth  coil,  etc.,  so 
that  coil  8  is  the  coil  nearly  opposite  to  that  in  which  resistance  1 
is  inserted.  Evidently  coils  9-16  would  give  results  similar  to 
those  obtained  from  1-8  and  so  the  curves  are  not  given  here. 


APPENDIX 


245 


In  Plates  19-22  are  given  the  current  forms  of  every  other  coil, 
from  one  A.C.  tap  to  the  next  A.C.  tap.  In  spite  of  the  fact  that 
the  D.C.  current  output  is  not  uniform  (see  Plate  19)  and  the  low 
efficiency  of  the  rotary  used,  the  curves  obtained  are  strikingly 
similar  to  the  curves  obtained  theoretically. 


The  field  forms  of  the  auxiliary  pole  rotary  were  examined  by 
taking  oscillograms  from  a  search  coil  made  up  of  10  turns  of  fine 
wire  laced  into  two  slots  spaced  180°  apart.  This  coil  was  con- 
nected to  a  couple  of  extra  slip  rings  mounted  on  the  shaft  of  the 


246  APPENDIX 

rotary.  Plate  24  gives  the  field  form  when  the  main  pole  only  was 
excited;  Plate  25  that  produced  by  the  main  pole  and  auxiliary 
pole  both  carrying  the  same  current  and  both  poles,  main  and 
auxiliary,  magnetized  in  the  same  direction;  that  is,  the  main  pole 
and  its  adjacent  auxiliary  have  the  same  polarity.  Plate  26  shows 
the  field  form  when  the  two  poles  had  the  same  current  strength  as 
in  Plate  25,  but  the  current  through  the  auxiliary  pole  had  been 
reversed  so  that  its  polarity  was  opposite  to  that  of  its  adjacent 
main  pole.  It  will  be  noticed  that  although  the  auxiliary  pole 
carried  the  same  current  when  Plates  25  and  26  were  taken,  the 


strength  of  magnetic  field  under  the  auxiliary  pole  in  Plate  26  is 
less  than  that  shown  in  Plate  25.  This  is  due,  of  course,  to  the 
much  greater  leakage  flux  when  the  main  and  auxiliary  poles  have 
opposite  polarity  than  when  they  have  the  same  polarity. 

The  zero  lines'  of  the  three  curve  sheets  are  slightly  displaced 
but  it  will  be  noticed  that  with  the  fields  excited  with  opposite 
polarity,  as  in  Plate  26,  there  is  considerable  flux  at  the  point  of 
commutation  and  sparking  occurred  while  the  curves  were  being 
taken.  The  main  field  should  have  been  reduced  to  overcome 
this  difficulty. 

Plate  27.  The  forms  of  the  E.M.F.  waves  generated  between  the 
different  taps  of  a  regulating  (auxiliary)  pole  rotary  were  examined 
by  the  ondograph  for  various  conditions  of  field  excitation.  Curves 
1,  2  and  3  represent  the  wave  form  of  voltage  generated  between 
120°  taps ;  the  machine  was  being  run  as  an  inverted  rotary,  un- 


APPENDIX  247 

loaded,  when  these  curves  were  taken.  Curves  4,  5  and  6  are  cor- 
responding curves  taken  between  180°  taps.  Curves  2  and  5  were 
taken  with  main  field  only  excited.  Curves  3  and  4  were  taken  with 
the  auxiliary  pole  excited  in  the  same  direction  as  the  main  field, 
while  curves  1  and  6  show  the  results  when  the  auxiliary  pole  had 
polarity  opposite  to  that  of  the  adjacent  main  pole.  It  will  be 
noticed  that  the  E.M.F.  generated  between  the  180°  taps  suffers 


quite  a  large  distortion  as  the  field  distribution  is  altered  but  that 
the  voltage  between  120°  taps  retained,  under  all  three  conditions, 
nearly  a  sinusoidal  wave  form. 

There  is  much  material  for  analysis  in  this  plate  of  wave  forms 
but  it  is  not  thought  well  to  introduce  it  at  this  point.  It  will 
be  remarked,  however,  that  a  field  giving  a  wave  form  as  shown 
in  curve  6,  would  be  subject  to  very  great  changes  if  the  rotary 
were  run  direct  (i.e.,  A.C.  to  D.C.)  and  the  impressed  E.M.F.  was 
a  sine  wave. 


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